The neurodiversity movement weaved together with the American social justice movement and developed an extremist part. By 2013, some neurodiversity advocates wrote that autism and ADHD are the result of normal natural variation in the human genome. By 2019, some said that the real problem is the dominant ableist culture of society that sets standards for what it means to be normal. Autismist is someone who believes that tipp... Is that really, like, radical? I don't know if she used that particular term, but I wouldn't say that that's necessarily radical. I mean, it's pretty in keeping with the social model of disability. Not sure about that characterization. Some of them have put the idea of treating autism on the same level as the idea of treating homosexuality. For example, autism speaks as a charitable organisation with the stated goal of creating an inclusive world for all individuals with autism throughout their lifespan. They partnered with Google on a project called Missing to develop a database to identify genetic variants of autism. On their website, they described the project's mission as to identify many subtypes of autism with the goal of developing more personalised and effective treatments. Two years ago, the project was decried by some people on Twitter as Eugenics, the Nazi doctrine of cleaning society from genetic-clean inferior groups, which back then included disabled and autistic people. It seems to me that these tweets attacking the Missing project were severely misrepresenting what the project website stated. Nevertheless, Autism Speaks, which I remind you is a charitable organisation that supports autistic people, has been labelled a hate group by some activists. This extreme position seems to be held by a small group that is vocal on Twitter and I don't think it's representative for the neurodiversity community as a whole, but it's clearly a sentiment that exists. I wouldn't say that that position is necessarily extremist. You only have to do a bit of digging into it to understand it. Is Sabine saying that she's for Eugenics? I know she's saying that that's not the aim of the study, but... Now this is an interesting one, because I've never seen this person before. They have about 1.3 million subscribers. This is Why Is Everybody Suddenly Neurodivergent by Sabine Hossenfelder. So this might be quite an interesting one. Maybe an outside perspective, maybe a psychological medical perspective on it, possibly. Righty, let's go for it, guys. And three, Simon Baron Cohen, a clinical psychologist at the University of Cambridge, claimed that Albert Einstein had autism. Elon Musk has had his Asperger syndrome, and if you ask Google, you'll find many lists of famous people who supposedly have autism, usually with the preamble, though he was never diagnosed. It's been attributed to Mark Zuckerberg, Steve Jobs, Bill Gates and Eminem, none of whom ever said anything to deny or confirm it. What's going on? Why does everyone suddenly have autism? How is it different from Asperger's syndrome? What does neurodivergent mean? And what is internalised ableism? Am I autistic? That's what we'll talk about today. Science without the gobbledygook. Asperger's syndrome, like, man, it's a bit outdated. Like, it may be ASD1, like, if you're going to be going to take the science perspective on it. Although I don't know about the ICD-10. I don't know if that's still part of the ICD-10. Might be. First things first, what is autism? The term autism was introduced in 1911 by the Swiss psychiatrist Eugen Bloiler. He used to describe what he believed to be a childhood version of schizophrenia. The term autism alludes to the Greek word autus, which means self, a word that Bloiler used because the children seem to be detached from reality and withdrawn into their inner world, even though they didn't have smartphones back then. Autism was later recognised as a condition independent from schizophrenia. Today, the diagnosis of autism... I think that there is also a couple of extra parts to the history of autism. It's kind of the way that she's kind of going over it. It's like, this is where it came from. It's like, it was actually characterised by, like, something, something sucrava, I think, like, I think German scientists or something like that, who really kind of popularised and looked into childhood schizophrenia, which is kind of like, as she said, like a part of history in terms of how autism came to be. There was also another few prime movers in that way. There was Kana, who invented that kind of Kana's autism thing, which it was kind of a little bit of like a marketing kind of money-making thing for him because he made the criteria so narrow that it was basically just to, like, attracts, like, wealthy white people to, like, come get a diagnosis because that was a part of it. And then you obviously have Asperger's, Hans Asperger's who's, you know, has mixed feelings in the community about, you know, obviously, like, a part of the whole World War II thing. So there's different, like, people involved in that. I think Lorna Wing was also a really big prime mover in terms of understanding autism a bit better. She invented that triad of impairments, which we use a lot to explain, like, the traits of autism as far as, like, the medical and science fields go. Includes learning difficulties, especially with language and speech, trouble with verbal and non-verbal communication, avoiding eye contact, repetitive movements, specialized and often obsessive interests or behaviors, difficulties with emotional control and extreme reactions to stimuli, such as light, touch or noise. Autistic people also sometimes have unusual abilities, such as a remarkable memory for facts, numbers of visual details or doing mental arithmetic. This combination of cognitive problems... It's quite a rare thing. I think what she's trying to characterize there is fontism. It's quite a rare thing. Like, you know, someone who would be considered to be ASD-free, you know, very high support needs, just has, like, one skill that they're just exceptional at. It's very raffling. Blimps and abilities is the reason why some autistic people were once regarded idioservants. However, the word idiot has drifted in meaning since then and is now considered an insult, so the condition has been renamed to Sevant syndrome. Most people had probably never heard of autism until Dustin Hoffman introduced it to the world in 1988 with the movie Rain Man. The movie was inspired by Kim Peek, one of the most famous severs ever. But plot twist, why Peek had originally been diagnosed with autism in 2008, a group of doctors studied MRI images of his brain and concluded that he wasn't autistic, but most likely had FG syndrome, a genetic condition linked to abnormalities of the X chromosome. In 1943, the Austrian pediatrician Hans Asperger described a group of children who had difficulties with social interactions and communication, like autistic people, but who did not have the same troubles with language and cognitive function. This high-functioning autism became later called Asperger syndrome. Fast forward 70 years and psychologists decided that there is no clear distinction between Asperger syndrome and autism, and the term high-functioning autism is neither useful nor descriptive. Instead, there is an entire spectrum of symptoms with expressions that greatly vary. This is why the term Asperger syndrome is no longer in use today. It's now called autism spectrum disorder, ASD for short. ASD subsumes both Asperger syndrome, autism, and some other developmental disorders. Why Elon Musk referred to a retired expression like Asperger's, is anyone's guess? Maybe he spends too much time with this old people app called Twitter. I'd say it's probably because most people, when you say autism, think of people who have extremely high support needs.  Because it's going against what they have in the head about what an autistic person is like. And so Asperger's, it can sometimes be a little bit more like friendly to the public in terms of them understanding what you're talking about and not making like assumptions or like thinking that you're lying or like, you know, stuff like that. I think that's probably the reason why. And also like, perhaps Asperger's might not have as much stigma behind it than autism, not sure. Just giving it a guess.  I mean, say what you want about Rain Man. I mean, it's obviously like really like outdated in terms of autism depictions, but it was one of the first things that came out about it. It was kind of like a, I guess, like a start for autism being understood in the mainstream, I guess. Besides the previously mentioned difficulties with language, social interactions, emotional control and repetitive behavior, studies have found that ASD is correlated with anxiety, sleep problems, seizures and an elevated risk of gastrointestinal problems. Symptoms of ASD are usually diagnosed in early childhood and persist throughout life, though they can improve or get worse. The spectrum of the disorder does not refer to the strengths of the symptoms, but rather their variety. How strong the symptoms are is measured separately by three different levels, which indicate how much support the person needs. One is the lowest level, three, the highest. Of course, no one likes to use a clunky term like autism spectrum disorder, which is why most people still refer to it as just autism. To avoid confusion, what was previously called autism is now called classical autism. The other expression that has taken hold for having autism spectrum disorder is being on the spectrum, though I... Classical autism really loves to play the piano. Classical autism.  I don't think they should have called it quantum autism. Several studies have shown that the brains of children with ASD develop differently than those of most children, and that this difference is often visible in brain scans before the onset of symptoms. For example, a 2017 paper in Nature found that in ASD children, some parts of the brain grow noticeably faster than average, and that this growth is linked to the severity of social difficulties. Other studies... That's interesting. I know that there are some studies, perhaps going into development of the brain to do with neuronal pruning, where the connections are kind of pruned, not doing a very good job of explaining that. You've never watched Rain Man? I'd give it a watch, you know? It's definitely interesting. It's good to, I guess, see why people don't like it.  I did enjoy watching it. If I was to view it in terms of its accuracy and its impact on the autistic community, then I probably wouldn't have enjoyed it. But I thought it was somewhat touching in certain ways, and I feel it is kind of somewhat overshadowed by this extreme savantist kind of skill that's very, very rare. It's just not really very representative. And a lot of people talk about it when you mention autism. They're like, oh, Rain Man, they're like, no. That one's not a Rain Man. These have shown that compared to the average person, people with ASD have significantly fewer connections between parts of the brain that are used for social interactions, and instead, more local connections in regions associated with sensory control. These properties can be inferred from brain scans before the age of two. How common is autism spectrum disorder? The worldwide prevalence of ASD is about 1%, but in high income countries, some estimates say the rate is higher. For example, a 2018 review by researchers from the UK found that ASD prevalence in England and North America is between 2.4 and almost 10% depending on region. And another factor to consider into this as well is masking on undiagnosed autism. So who knows what the percentage is like. It could be very, very high. It could be somewhat like around the 1% to 2% bull park. Who knows? According to the American Center for Disease Control, about 1 in 36 children in the United States were diagnosed with ASD in 2020. Probably quite a few of you who are watching this video are autistic. I'm afraid I have nothing whatsoever new to tell you. The number for ASD diagnoses has increased substantially over the past. Okay. We're done now. There's nothing new to just cutting out the autistic people from the video. I'm sure I think you're underplaying it, Sabine. Come on, give us some new factual information. Two decades. In 2000, it was only one in 150 children who were diagnosed in the US. This means the number has increased by almost a factor of five in 20 years. The increase in the UK has been even faster, almost a factor eight in the same period. That made- And this is not necessarily this increase, a result of autism becoming more prevalent, but the increased frequency of diagnosis, which is definitely like a good thing. So you're just catching more autistic people, basically. Not saying that that could be the only thing. Maybe the prevalence is increasing. Who knows? But I'd say that's a really big factor involved in this increase. Understanding de-stigmatization, more education around it, more research, more light input from the autistic community. The major reason for this is higher awareness and better screening. At the same time, the fraction of those severely affected by ASD is dropping. In the 1980s, more than two-thirds of people diagnosed with autism also had an intellectual disability. By 2018, the fraction of ASD children in the US classified with an intellectual disability, that's an IQ below 70, was 31%. This decrease of the fraction of those with severe disability is also likely due to the enhanced awareness and screening, which picks up on the less obvious cases. Yeah, picks up on the ASD ones.  At the moment, there are about four times more men than women diagnosed with ASD. The reasons for this are not well understood. Several studies have shown that young girls are less likely to be diagnosed with ASD because they're better at hiding symptoms, a strategy that's referred to as masking. Masking includes copying facial expressions and social behavior or memorizing a repertoire of answers to questions that'll be accepted as normal enough. While masking might help girls with social integration, it can cause other mental health problems such as depression, anxiety, and low self-esteem. In later life, ASD in girls might get covered up by conditions such as eating disorders or OCD. Girls might also be more successful at masking because their social environment makes it easier. For example, a 2017 paper by American researchers observed 96 elementary school children, half of which with ASD. They found that ASD girls weaved in and out of social activities in loose groups, making it hard for anyone to tell whether they integrated well. Boys, on the other hand, tended to play organized games and the ASD boys just played alone so they stood out much more obviously. That's very interesting because that characterization of autistic girls, that's pretty much my experience in secondary school, social drifting between integrating within different social groups. That's definitely why I used to do. But also, I feel like even when I was younger, when I was a boy, when I was a child, a young boy, I didn't to enjoy being a part of group games. Not gonna lie. I think the interesting thing about the boy-to-girl ratio as well, it's definitely something that I've seen. I've seen a lot more undiagnosed women than undiagnosed men in my life, just going out through life. Yeah, I mean, mirroring is a component of masking. It's a component of social camouflage. You have the social camouflage being, I've got a video coming out on it actually tomorrow, a full presentation-style video, which social camouflage is split into assimilation, compensation, masking. Mirroring fits under one of those other categories rather than masking, but it's definitely a part of social camouflage. How much of the sex difference in ASD diagnosis is due to masking is presently unclear. What is clear, however, is that diagnosing girls requires other methods than those used for boys. And also, women definitely do tend to get more of the diagnosis of BPD than men do. Heard a lot about that from testimonials and also some statistics around it as well. BPD is very common misdiagnosis. ASD is normally diagnosed in childhood below the age of five, especially now that doctors look for symptoms during regular childhood screening. However, this is a fairly recent development. The increasing awareness of the condition has also resulted in larger numbers of people being diagnosed later in life because they fell through the cracks as children. Still, it can take a long time, sometimes years to get a diagnosis. It's a problem because the lack of diagnosis can make it more difficult to cope. A study done in the UK estimated that 10% of suicide victims had signs of undiagnosed autism. What causes it? Classical autism was originally believed to be a psychological condition that results from poor parenting. This idea was pushed in particular. Refrigerator, mother, something that was pushed out a lot. You know, I'm very empathetic towards like autistic parents as well because they've had their own wrap, like own like dealings of stigma around autism too. You know, it's a lot of people blaming mothers who are not like warm enough and not communicative and physically, like physical enough with their kids and so they develop autism. It's complete, complete box. Nuclear by the psychiatrist, Leo Kanna. He argued that the leading cause of autism was the refrigerator mother who didn't love her child as she should have. We now know that ASD is not the fault of maternal temperatures, but an unusual variation of neurodevelopment with a strong hereditary component. That is, children of people with ASD are considerably more likely to also have it. Different studies have put the probability at about 40 to 80%. Part of the difficulty of pinning the number down is that ASD has so many different facets. There are loads of genes involved and quite possibly some aspects of it are more others less hereditary. Some environmental factors also seem to play a role. For example, some studies have found a link between obesity or diabetes of the mother and ASD of the child. Really? Really? Odds relative risk ratio. Neonatal hypoxia, gestational diabetes. Valproate. Um, maternal age, paternal age, paternal age as well. Oldest symptom of ASD, really high. Preterm birth, paternal obesity. And it reduces with folic acid intake. Interesting, very interesting. 22 study as well, 2020. Wow, I think as well that there is, if I'm not wrong, some correlation between like polycystic ovary. I can't remember the actual term. Polycystic ovary syndrome, I think. And autism, if I'm not wrong. Though the increase in the odds ratio is not large and as it's often the case with studies like this, it's difficult to infer causation from a correlation because there could be underlying causes linking both. 2016 meta-analysis by a group of Chinese researchers looked at the link between the age of the parents and the chance of the child to have ASD. They took into account 27 studies with a total of almost 67,000 cases and found a clear correlation, especially with the age of the father. Per every increase of 10 years in maternal and paternal age, they found an associated 18% and 21% higher risk of ASD, respectively. You see, this is interesting. Again, as she said, this is a correlation. There might be, I might have to look into it a bit more but could be wrong not saying that this is at all factual but I feel like a lot of autistic people that I speak to, they don't tend to be that focused on having children as opposed to a lot of neurotypicals that I've spoken to. So maybe the increase in age of having a child might just be a fact that the parents were autistic, maybe. It's just a suggestion. I definitely have to look into it a bit more but some made that little little connection in my brain completely based on subjective experience and observing other people but that maybe could be something. Definitely need to look into that a bit more. We've also learned about a few things that do not cause it other than cold months. For example, vaccines. In 1998, an anti-vaccine activist managed to publish a fraudulent paper in the Lancet claiming that measles vaccines cause autism. He was struck off the medical register and the paper was retracted. If you're watching this channel, you probably don't need to be told but vaccines don't cause autism. They don't cause autism spectrum disorder either. To name just one example, a 2014 meta-analysis summarized 10 different studies and found no evidence of a relation between vaccination and ASD. Other things that don't cause ASD are caesarean sections in vitro fertilization, plastic, wrapper, bacterial infections and watching YouTube. ASD is the lifelike. Oh, that's kind of like, she's got this, this, this pizzazz, this like tongue-in-cheek humor. It's just delivered very like bluntly. It just gets me. Like, I love that style of humor. It's so funny. Oh my God. Yeah, vaccines. Yeah.  It's, it's, it's bull, like, and it's dangerous, 100%. Long condition. There's no cure and there's no medication. However, early diagnosis and intervention with suitably targeted programs can greatly improve life satisfaction and medication as well as psychotherapy can be used to deal with some of the symptoms. And some of these interventions also have some, depending on what intervention you're going for and if you're autistic and you're in the autistic community, you know, I'm talking about, can also cause really negative lifelong things as well. Currently, the treatment focus is on psychological and social support for ASD children and their families. Some young autistic people don't speak at all and can benefit from using picture cards or speech-generating devices. Targeted programs that are started as early as possible can improve attention, language development, social engagement, and can reduce the severity of symptoms later in life. The most successful of these programs are those which include parents and schools because much of it comes down to other people understanding triggers and trying to avoid them. Really, really happy that it's been brought this up because as someone who's very, who applies and, you know, believes a lot in the social model of disability, I think this is a really, really important aspect to it. 100% in schools, that is where we need it. That is where a lot of the mental health comes from in my eyes, just from looking at the research. Some of the things that we experience in schools can have real lifelong impacts on our overall mental health well-being. For example, children with ASD might be bad at rapidly switching between tasks or be unable to read non-verbal clues, both of which can be avoided to a large degree. They also often use repetitive motion such as rocking or hand-flapping to calm themselves down. A behavior referred to as stimming. Asking them to stop is exactly the wrong thing to do. Those are some examples where awareness, among other people, can really make a big difference. There are also some medications to treat co-occurring symptoms like irritability, agitation, anxiety or depression, and some might benefit from cognitive behavioral therapy. However, there's no one-size-fits-all approach because the symptoms of ASD are so varied. According to the diagnostic and statistical- A lot of the medications that she brought up there were very much like ADHD-related things. Methylphenidate, you know, Ritalin, all that stuff.  Autism spectrum disorder is a mental health disorder. However, a lot of people with ASD and other conditions that have traditionally been labelled disorders feel that description is inappropriate. Just because they're not typical doesn't mean there's something wrong with them. The term that many of them prefer is that they are neurodiverse, as opposed to neurotypical. Sabine, you are like the golden child. I was so ready to add in a little bit of the autism community angle to stuff.  The term neurodiverse refers to a group, whereas an individual would be described as neurodivergent. Neurodiversity includes not just ASD, but also attention deficit hyperactivity disorder, dyslexia, and a few other conditions. That's something that she's got wrong. This is something that I got wrong as well, but neurodiversity is not a restrictive term. It kind of makes it a bit more difficult because, like, in my eyes, at some point, the idea of neurodiversity was sort of like a deviation from that kind of viewing it as a disorder, viewing it as something that's the wholly negative. It was kind of like, okay, it's not all bad. It can actually be positive and can be neutral and et cetera. But, you know, the actual definition of neurodiversity is the variation in the human brain and that covers everything. Even some things that I would probably say are completely negative, like, you know, there's some cognitive things that are just flat out, just not good, like, for the person.  That's all covered under neurodiversity. But I like the characterization that she's using. I very much do. But it's not correct. The term neurodiversity was coined in the late 1990s by the Australian sociologist Judy Singer, loosely based on the expression biodiversity. She saw it as an extension of social justice movements that had given women and homosexuals rights they should have had all along. Singer argued that neurodivergent people had not been given their rightful place in society either, and it was about time to stop pathologizing them. The neurodiversity movement weaved together with the American social justice movement and developed an extremist part. By 2013, some neurodiversity advocates wrote that autism and ADHD are the result of normal natural variation in the human genome. By 2019, some said that the real problem is the dominant ableist culture of society that sets standards for what it means to be normal. Enableist is someone who believes that typically... Is that really, like, radical? I don't know if she used that particular term, but I wouldn't say that that's necessarily radical. I mean, it's pretty in keeping with the social model of disability. Not sure about that characterization. Colabilities are superior. If you're neurodivergent but have internalized ableism, you may mistakenly blame yourself for the challenges you face when, in fact, the problem lies with the societal barriers and discrimination that make it harder for neurodivergent people to thrive. Some of them have put the idea of treating autism on the same level as the idea of treating homosexuality. For example, Autism Speaks is a charitable organization with the stated goal of creating an inclusive world for all individuals with autism throughout their lifespan. They partnered with Google on a project called Missing to develop a database to identify genetic variants of autism. On their website, they described the project's mission as to identify many subtypes of autism with the goal of developing more personalized and effective treatments. Two years ago, the project was decried by some people on Twitter as eugenics, the Nazi doctrine of cleaning society from genetically inferior groups which back then included disabled and autistic people. It seems to me that these tweets attacking the Missing project were severely misrepresenting what the project website stated.  This extreme position seems to be held by a small group that is vocal on Twitter, and I don't think it's representative for the neurodiversity community as a whole, but it's clearly a sentiment that exists. I wouldn't say that that position is necessarily extremist. I mean, you only have to do a bit of digging into it to understand it. Is Sabine saying that she's for eugenics? I mean, I know she's saying that, okay, that that's not the aim of the study, but I think the issue when it comes to, like, I was actually, there was a subsection of this in the UK called the Spectrum 10K, which I believe was somewhat affiliated in that regard, and that was shut down in the UK as well. I actually interviewed Simon Baron Cohen and released my episode with him on the day of, I think, the Spectrum 10K being released. You know, I saw the utility in it. You know, I think that it could have been a really good thing because if we could rule out the sort of the genetic causes of autistic people experiencing mental health and actually point to the fact that it's our environment then perhaps there'll be a lot more evidence to put towards actually making some improvements to improve the life quality of autistic people. That was my view on it. I think the reason why people are against this idea of genetically identifying genes related to autism is that it can be used in other ways. And perhaps it may not be used in the UK because there is a lot of people who do not like this organization or like this sentiment of sort of curing and preventing autism. The thing is is that it could be used in other ways. And there is a probability that it could occur, like even in the UK, it could occur. I mean, it happens with Down syndrome, perhaps. Like people can sort of test for Down syndrome in the womb and choose whether to keep that embryo or not. What ones if that happens with autism? What ones if people have the ability to test and say, okay, I don't want an autistic kid? And that is the eugenics like looming over that aspect of things. And obviously people are going to have their own opinions of it. I don't think that it's a good thing that would happen. I also think that there would be other countries which are involved in this whole kind of open source study who could use that within their own population. People outside of the UK and the US, Australia, perhaps. That could be something that happens. So I don't think it's right, saying that it's a kind of like a radical kind of view because it's from what I've learned from the autistic communities talking to over autistic people. It's generally a bad sentiment to promote eugenics. And there is a lot of background to this organization that's been put forward. I wouldn't necessarily say that it's a hate group. I think that's a bit like over the top. But it definitely hasn't had the intention of putting resources into supporting families necessarily. It can be a very good source of information, some sort of medical information that does come up a lot on Google, which I see the utility of. But again, it was made in order to cure autism in the start. And I don't think that there's been sort of an accountability there of saying like, okay, this was actually how we are and we're trying to change. And it really hasn't been that long since they changed their sort of motives towards it. And I think that's why a lot of people are very hesitant to do so. It's a point of view. I wouldn't say that it's a radical point of view. This whole idea of ableism is also very real. Internalized ableism as well, making yourself feel bad for something that you can't control necessarily. I don't necessarily think it's always about over people. It could be comparing yourself to people who aren't autistic and sort of feeling bad about not being able to reach those standards or be that kind of person or do that kind of thing. It's kind of going a little bit more into like the social side than I expected it to. But let's continue. Let's see what else she has to say. Fast forward to 2023 and TikTok is full of teenagers celebrating their neurodiversity, some of them self-diagnosed. Now you might say there's nothing wrong with teenagers wanting to feel good about themselves. And I fully agree. The neurodiversity movement has a point. Our world is built for typical people and typical people are often not mindful of those who are less typical. In most cases, I think not so much because typical people are actively mean but passively careless. If you're too tall or too short, too loud or too quiet or too anything really, you'll have trouble fitting in. Some people's brains don't work like yours but that doesn't mean they are the problem. The problem might be that you're not making the necessary effort trying to understand them. Paying attention to those outside a standard deviation of the average makes their lives easier and enriches our society. That sounds all well and good. Problem is that most neurodivergent people you see on TikTok are those well enough to produce TikToks. And in contrast to being queer or female, ASD can in severe cases significantly impair person's ability to live independently. This is why the neurodiversity movement has seen somewhat of a backlash in recent years. Primarily from caregivers of people with level three ASD who feel like talking about internalized ableism doesn't help. For example, the London Bay... Very, very mixed feelings on this video now. I very much empathize with parents of people who do have ASD-free children. I think sometimes, I think a lot of the time actually people within the autistic community are kind of neglect to consider that. I do also find that people who do, who are characterized as being ASD-free, also come with an intellectual disability. I would argue the intellectual disability element that is the disabling part to a very large degree. I don't think it's always like that. So it's a bit of a gray area for me. The thing is with... I don't know, the way that she's framing it, it's kind of like... It doesn't sit right. I don't think she's kind of grasping the whole of it. A neurobiologist, Moab Constande, has written an article for Eon titled Against Neurodiversity in which he draws attention to a worrying trend of romanticizing autism that has extended to other conditions that can be severe, debilitating and life-threatening. Yeah, I think that's also like, of course, you're going to have the fringe cases. You're going to have the radical aspect of it. You know, what you can consider something... I've talked about before with psychopathy, which is a genetic thing. And psychopathy, which is more of an acquired neurodiversity, if you want to call it. Man, there is nothing good about having seizures, being epileptic. But the thing is, she's coming from the standpoint of neurodiversity being kind of like a counter-movement to autism being seen as a disability, whereas that's not how people characterize it within the circles that I've been a part of, particularly. It's more characterized as like... Yeah, a lot of people say it's neurodiversity and a disability. So it's not necessarily this kind of like contrasted to perspectives on it. It's more or less kind of drawing attention to the realities of it, some of the actual neutral and positive things that are involved within it. It's not all this kind of negative thing. And the reason why it's not all this negative thing is because it's true. It's not all negative. There are actually some really good things about being autistic. But I also don't think we should shut down parents of autistic people who do have children or the voices of people who are like quite intellectually disabled. It is an important component to think about, of 100%. Such as depression and schizophrenia. He writes that the idea that autism is a variation of normal is at odds with scientific understanding of the condition and that in their zealous pursuit of autistic rights some advocates have become authoritarian and militant, harassing and bullying anyone who dares to portray autism negatively or expresses a desire for treatment or cure. Yes, this happens. I think this happens in any type of social, socially like focused community. Like you're going to get people like that. I'm not one of those people. However, I do express distaste for the idea of treating or curing my autism or being autistic. Yeah, it's a tough one. But I, eugenics, man, like, come on. Tom Clements, who is autistic himself, wrote in an opinion piece of The Guardian that many now self-identify as autistic as though autism were a fashion label rather than a debilitating disorder. Oh my God. I get it. She's done a lot of research into it. She's seen it from both sides. It just feels like she was very much like, it feels like she's siding with some of these kind of mainstream opinions. You know, people saying that, oh, I'm autistic as a fashion label. Maybe like kids, children, people with other neurodivergent conditions, neurodivergencies, whatever, could be at risk for kind of misdiagnosing themselves. But the majority of autistic people that I've come across and know, they spend a long time contemplating learning about it. So it's definitely not like this isn't representative of the majority of adults, I would say. And that such an attitude has led to the marginalization of autistic people who, by virtue of their disability, are unable to speak and rely on others to do so on their behalf. Some neurodivergent people have pushed back, pointing out that Clements used the term high functioning to refer to himself and that such functioning labels should not be used because they suggest some people with ASD do not need help. Others complained that Clements is a very active show who spends a lot of time misrepresenting the neurodiversity movement and that he accuses autistic people of not actually being autistic. I didn't know anything about this. When I started making this video, I was just trying to understand the symptoms of autism and had no idea the topic had become so controversial. But I hope that this rundown helps you make a little sense of what's going on. In summary, autism spectrum disorder is a mental health condition that subsumes what was previously called autism and Asperger syndrome. It's more common than you might think affecting more than one in a hundred people. Most of them are able to live a fairly normal life but face challenges especially at work and in social interactions. Many of them prefer to refer to themselves as neurodivergent and try to raise awareness for the challenges they are facing. But the neurodiversity movement has been criticized for trivializing the problems of those most severely affected. I've been accused mostly by parents. And there is that really big divide between autistic adults and parents. I've seen it, it's very much like a thing. There needs to be more conversations about it. I've actually reached out to Autism Speaks for an interview because I want to talk to them about it. They don't want to talk to me. Like, I've tried, it's something that I want to do. I interviewed someone. Some of the videos are actually within this video that we're watching called the Autism Cafe who is a part of Autism Speaks. And has been subject to a lot of bullying actually from autistic people, which I'm not a fan of. She does also talk about the utility of ABA. So it's a good episode to watch if you want to I suppose understand kind of like our contrasting views on things or at least like hear her side of things, I guess. You know, I feel like a lot of these kind of like these news sources, pieces of information that they're very, they kind of paint it as a lot more of a radical thing than it is in reality, which it makes sense because social media outlets, they tend to focus on the loudest people, you know, and the loudest people tend to be like the most radical people. You know, I understand that it's just we need to get a little bit more like education out there about what it is that we're trying to do by, you know, like the actual autistic community. And a lot of people actually deviate a lot on their opinions and views, I would say. Let's see what she's got to say for the roundup and then we'll give a sort of a roundup to you.  Berger's syndrome. It normally comes after a long list of complaints about how I'm rude and arrogant, talk like a robot. I'm about as empathetic as a brick and similarly spontaneous, but considerably less social. Luckily, there's an online self-test you can do for this. So let's see how neurodivergent I am. Here we go. Your autism spectrum symptoms are high. I think that's wrong. I'm really just rude or German, but then I repeat myself. She scored higher than me. I have like mild signs of autism when I take those tests. But what she said about the German aspect of things, it's a lot of the diagnosis is very subjective and dependent on culture. So you could imagine if someone's from a different culture, like where different sort of social rules sort of dictate what the normal is. Makes sense that perhaps like it wouldn't be as sort of identified in certain cultures. What do you think about neurodiversity?  Let us know in the comments. I've often wished I could collect all news on one topic and see where it's been covered and who put what's been on it. And I've now found a way to do exactly this. It's a website and app called Ground News and they've been sponsoring this video. Okay, let's have a look at some of the comments. I'm interested to see like what people do. There's definitely a fine line between normalization and romanticization. I found it helps all to not consider my neurodivergent traits as weird, but it also feels odd to call them especially good. They are just, they just are and it's just me. Speaking for myself, I did not suddenly become autistic. I was autistic 58 years before I was diagnosed as well as, not everyone is neurodivergent. As a person who is not on Twitter, I want to say it. I think you might want to deeper dive into autism seeks before saying it's harmless. Sure, fearmongering ads, you could shrug off as just bad PR, but one of the co-founders, Bob Wright, has multiple occasions expressed his doubt on whether autism may be called by vaccines. All in all, this foundation is more interested in giving the parents of autism a platform rather than autistic people themselves. Yeah, and then you got, you got that kind of the fearmongering video that we watched when we had to look at Foster on the spectrum's video. You know, all in all, this video is a lot more comprehensive than I expected in terms of like the topics that she covered up, which is just going to talk about like the, the medical side of things. I was kind of like, that was kind of my approach, but it kind of turned into a little bit more of somewhat like kind of a social perspective, like looking at some of the things that are going on within the communities and the media. Yeah, it's not sure how I felt about it. I don't know what Sabine's personal thoughts are on it, but I definitely agree with one of the comments. Like she does need to do a little bit more reading into this stuff, perhaps like interview some autistic people, see what they think, get their voices like actually displayed within the video as well. I think that that could be something that's quite important.


All right, so today I'm going to be talking about a brief history of data analytics. And really, you can tell a brief history of data analytics if you don't know about data. So really, this can be said to be a brief history of data itself, not just data analytics. So where did we start getting data? The earliest example we have of human storing and analyzing data are the talistics. The Ishango bone was discovered in 1960 in what is now Uganda. And it's thought to be one of the earliest pieces of evidence of prehistoric data storage. Now, Paleolithic period or Paleolithic tribes, tribe people used to mark notches into sticks and bones to keep track of trading activities or supplies. They would compare these sticks and then they would know, OK, is our harvest going to be, do I have enough supplies or stuff like that? But really, the first calculator in the world was the abacus, the first dedicated device constructed specifically to perform calculations. And this was in Babylon, the first libraries where the first libraries kind of occurred. But the abacus, really, in modern times, most people don't know how to use it. But there are special breed of people in Japan that actually use the abacus for mental arithmetic, kind of polishes their mental or kind of critical thinking skills. Let me just show you something. Just watch this. Thank you. 9348, 4946, 1377, 450. Thank you. Thank you. So moving on from the historic times, let's talk about emergence of statistics. And here we have talked about a fellow called John Grunt in 1963, invented or first recorded experiment of statistic data analysis, recorded information about mortality. Then we'll move to 1865. The term business intelligence is used for the first time by Richard Miller Stevens in his encyclopedia of commercial and business anecdotes. However, he didn't mention business intelligence to refer to BI technology that is being used today. What he did was Devons used it to describe Sir Henry Fornes' strategy to make profit. So Sir Henry Fornes, a banker, was able to gain advantage over his competitors by collecting studying data, or acting on data that he collected. But really, when we also want to talk about the early days of modern data storage, we need to talk about Nikola Tesla. Now Tesla is known for other things, but Nikola Tesla states that in an interview that when wireless technology is perfected or perfectly applied to the whole earth will be converted into a huge brain, which in fact it is. All things being particles of a real and rhythmic whole. That seems to be kind of interesting. But he said that the telephone we know today and in the future is going to be as small as being able to put in your pocket. That's what in 1926 he made that prediction. In 1928 Fritz Pulmer, a German Austrian engineer, invented a method for storing information magnetically on tape. The principle he developed is still used today with a vast majority of digital data being stored magnetically on computer hard disks. Now unfortunately, this is where I stop on my presentation and the whole story will continue up to today. But yeah, this is just these 12 slides. So thank you very much guys for listening to the abridged version. And what do you think?


Yesterday, I told you about a little competition I was having for the triangle inequality that I talked a little bit about yesterday, I'll talk about it again today, to confine the nicest proof. And there was a nice proof sent to me last night that maybe I'll tell you about, because it's rather nice. Reminded me of another competition, well, it sounds lecture also. And the Americans here will know what I'm talking about. Many of you may know of a Putnam competition, one of these competitions where people have to answer clever and witty questions in a time period to show how clever they are. And it's a national competition in North America, and there's some fun questions there. And I happened to be chair of the committee setting the exam in 2002. And one of the goals in all of these everywhere in the world is you put the year into one of the questions, right? And we're really struggling as a committee to find a question with 2002 in there. And this was just soon after all these conjectures that Sam talked about about moments had been made. And I happened to have a post-doc, Nathan Inge, at the time, who was into all these moment conjectures. And I said to Nathan, you've got to help. Is there any question you could think of that leads to 2002? Nathan came back with, well, it's this divided by 12. You try and find a problem that is suitable for the partner matter. We had fun for two days, but it was always going to be a little too hard for the partner. And that's the new competition. Find a great problem involving the eighth moment that gets 2002 out of there. OK. I'll try and be a little less silly. So yesterday we got to looking at this function. I told you that the so-called pretentious R-H, I'll write it over here, says that for all s, such that the real part of s is greater than 1, for all into just k, the k-th derivative over k factorial is less than less than something depending on s 2 to the k. Or let's just give something explicit, say 1 plus t to the epsilon, where s is sigma plus i t. OK. As I said, conjecture could be any constant that depends on s, but, and we saw yesterday this implies the Riemann hypothesis, and I claim was the starting point for what we're going to do. So let's just quickly run this through Perron's formula, which we wrote up yesterday. So as is the Dirichlet series for the sum of a n over n to the s. So all I really have to do is read off the n-th coefficients to interpret this formula. So what are the n-th coefficients to sum n less than or equal to x of the n-th coefficients? Well I'll just put as approximately 1 over 2i pi, the integral just to the right of n somewhere, here I want c greater than 1, of z of s ds. And zeta prime over zeta, the n-th coefficient is minus lambda n, so let me put a minus on this side. So we've got lambda n, so minus zeta prime over zeta, the n-th coefficient. The n-th coefficient here of minus zeta, the n-th coefficient will be minus 1, and this only has one coefficient which is right at the start, so, well, we can lose the two gamma in the approximately. So well this is psi of x minus x, and this sort of tells you, well it's good to think of lambda n minus 1, you've seen the sort of balancing act in Green-Tau, you sort of take off the main term and deal with the rest in various of their works, anyway, tends to work nicely in this. I do want an x to the s over s, how did you know, okay, so we want to play the game with case derivatives, so what happens when I take a case derivative, when I've got a Dirichlet series of a sum a n over n to the s, and I take the case derivative with respect to s, then all that really happens is I just get a minus log n each time I take a derivative. So actually if I just put a k up here, case derivative, there's a plus or a minus, I'm not really concerned which, and I've got a log n to the k here. So log n is, well as we talked about last time, I'm sure you know, you can go between psi x and pi x when thinking about the counterprimes very easily by partial summation because log is a very smooth and benign function, and as long as this moment isn't incredibly high, it's not going to make a lot of difference talking about lambda n minus 1 or lambda n minus 1 times log n to the power k. So okay, so let's do what we did last time, which is just to take absolute values and see what happens. So this right hand side then is less than, less than. Remember c is 1 plus 1 over log x which allows us to say that x to the s has absolute value x to the c which is e times x. So let's see, we're going to use this formula here, we're going to assume, so this is assuming the pretentious Riemann hypothesis, and we get, well forget the constant, so we get some sort of integral between minus t and t, so we take, make the change of variable c plus i t like last time. We've got this 2 to the k, k factorial, the 1 plus t to the epsilon is obviously less than t to the, let me actually, sorry, leave it as t to the epsilon, 1 plus t to the epsilon, sorry. I've got my x, I'll pull that outside. We've got this 1 plus t in the denominator, dt, okay, so that's an upper bound. Now let's, this integral, the only integrated bit is this part and that's when you integrate it, just going to be t to the epsilon, okay, because it's really t to the epsilon minus 1, and nothing bad happens from there, t equals 0. So this is less than, less than x times t to the epsilon times 2 to the k, k factorial, which is essentially 2 to the k, what, 2k over e to the k, well, there's times a root k, but that will go into the t to the epsilon, it's just using Stirling's formula. Okay, and there is, I probably need to take account of the error term to this correctly, my notes are a disaster, okay, there should be another log, but let me just write it as this for now, let's not worry too much, so what we want to do, yeah, okay, so we're eventually going to divide, you know, when we do partial summation there'll be a divide through by log x to the k, so what I want to do is, is what we will get from partial summation when we do it, is something like, and there may be a few logs that occur because of the partial summation, but we'll get something like x t to the epsilon 2k over e log x to the k, okay, so that log x to the k comes from here, and then all we have to do is make the optimal choice for k, and the optimal choice for k will be k is log x over 2, yeah, you always optimize these things like that, so when I take k equals log x over 2, so forget that, sorry, k equals log x over 2, oh, then I get less than less than x t to the epsilon over x to the half, and so x to the half t to the epsilon, so I take t to be around x or something and I'm in good shape, so we've again proved, well proved directly now as z of s, that making this assumption about this function implies the same estimate that you get from the Riemann hypothesis as we saw last time, and what I didn't say last time is that actually this is equivalent to the Riemann hypothesis, getting a bound of x to the half, so we'll just finish this off by saying less than x to the half plus epsilon, just choosing t, say as root x or something like that, okay, so that's using the pretentious Riemann hypothesis as we call it to prove a very strong estimate on psi x minus x, and now let's just see what happens with this if we plug in a different estimate, and what we're going to be able to prove instead of this is something like log t, log 1 plus t to the k, c, some constant log 1 plus t to the k, and I'll mention in a minute how to do that, so let's actually start from reminding ourselves what we did initially to prove that this shows the Riemann hypothesis, this conjecture, if we replace this now by this estimate, this is what we'll get unconditionally, and I'll sketch that in a minute, then what we have, so let's say s0, s0 feeds hypothesis, when we look at z, that we look at z of s close to s0, then what's the radius of convergence of z of s that we can get if we make the assumption that the derivatives are bounded this way? Well, again, if you're going to get terms that have s minus s0 to the k times this thing, right, so this is going to converge in something like s minus s0 is less than some constant over log 1 plus t, just, it's the same idea of a 2 to the k that we have that, and so in particular, this gives a zero free region, so this is exactly the classical zero free region of de la Vallée-Poussin that comes out of proving this estimate. A little more transparent, I didn't get to go through the proof yesterday of a zero free region, but if you go through it, it's a little opaque, how, I mean, you get used to it, but it's a weird proof. This is, I think, a lot easier to see that why this is zero free region, it's just you have a Taylor series, it obviously converges in this ball. So if we plug that estimate, well, we can go through a zero free region and then prove the usual prime number theorem, or we could just play with it in here, and then the only real change is that we change that to 2 to C log 1 plus t to the k, and so when we come down here, we have, what, k times C log 1 plus t over E log x to the k, that's, I mean, it's all, it's just replaced 2 by C log 1 plus t, and then the optimal thing will be k is log x over C log 1 plus t, and so we end up with less than, less than x to the 1 minus some constant over log 1 plus t. Okay, something like that. Well, we have to be careful, so maybe I should say this. I'm working out to be this. We have a second error term in the prime number theorem that's x over t, maybe times some powers of log. You, as normal, you're just optimized where the 1 over t is the same as the x, the C over log t, just like in the classical proof, and so that optimization occurs where log t is like root log x, and so you end up with the upper bound x over e to the C root log x. Okay, so that's a recovery of the classical one you've all done. They'll have a little proof with a little number 10 thrown in as I talked about last time. Yeah, so the issue to improve a prime number theorem, to improve the error term in the prime number theorem is to improve this bound, if we can. Oh, actually, I think I should prove this bound. Let me go ahead and do that. Any questions on that? I was very pleased to see that sound loss control in his lecture at the back. Okay, so how might we go about proving such a bound? So we need bounds on z of s was, remember, well, I've got it up there, zeta prime zeta plus zeta minus 2 gamma. When we take the kth derivative of z of s, then we're going to have the kth derivative of zeta prime over zeta k times plus the kth derivative of zeta. So these are things we're going to need bounds on. Well, this presumably is straightforward because this guy is just the sum of, well, we'll talk about this in a minute. I'm not suggesting it's trivial, but let's say it's doable with a plus minus. Okay, so that's the kth derivative of the root of zeta. Remember, we're always to the right of 1. This one is more tricky. What do you do with it? Well, when you differentiate zeta prime over zeta, you get, what, zeta prime prime over zeta minus zeta prime over zeta all squared. You do it again. You do it again. There's some combinatorial mess, and it just so happens that that's been sorted out for us. Here it is. And this is, I guess, due to, was it Fadi Brano, Bruno? So if you find a big mess like this and it's a combinatorics book and they claim it as a combinatorialist, it's usually from like 1730. If you find this in a computer science book and they claim it for one of theirs, it's usually from 1987. So we believe a combinatorist that Bruno was probably the first. Anyway, so Which is Phi. Phi to Bruno. Phi. What language does that last thing belong to? I'm supposed to know. You're the one who speaks all the languages. Okay, so zeta prime over zeta is a big old combinatorial mass times something involving zeta differentiated a certain number of times over zeta. That's the way to look at this.  So actually this is, for us, this is all that we really care about, 2 to the k minus 1 times the maximum of this last thing. Okay, taking absolute values. So you're just sort of taking the worst term and assuming every term's bad. Okay, so we're going to have to work with a lot of terms like zeta ks or zeta js over zeta s. So what do we need? We need upper bounds for these guys, for the zeta j's and we need a lower bound for, or an upper bound for 1 over zeta s which is a lower bound for zeta s. As you might imagine these are fairly standard things in the literature. This is particularly to the right of 1, to the right of 1 convergences on your side. But let's just take a brief look at what might be difficult. So if I want to just sum this up, and now I'm thinking about s being sigma plus i t. And again, it's not very convergent. So sigma's just to the right of 1. I want to sum this guy up, over all integers n. Then the obvious way to do it is to say, well, this is roughly the integral. Right? I mean we approximate sums by integrals. Even though this is complex, so we've, and we have to be, you know, if it's, if it's real this is an easy approximation, easy to get good error terms. If it's complex, that's a little harder. And essentially you want something, if this is going to be a correct approximation, you want things to go smoothly. Now let's just forget the sigma for a minute and look at the end to the i t. What happens, say, is you go between 2 to the i t and 3 to the i t, things are going to be smooth if there's roughly the same angle involved. Yeah? If the, the, the gradation by angle as you move from one into the next isn't much.  You can't really approximate by the integral. So if t is, say, 10, 2 to the i t in angle is nowhere near 3 to the i t. Right? But a thousand to the i t is very close to a thousand and one to the i t if t is 10. If you think about it, well, it just as I go from n to n plus one, let's just compare the angles, n plus one to the i t over n to the i t looks something like one plus what, i t over n if you look at the Taylor series. Right? So where things start to get okay is when n is bigger than t. And if n is less than t, things are not okay. This is not a good approximation. So essentially we have to break this up into two parts. We have to say this is like the sum from n equals one to n of one over n to the s plus the sum n greater than n of one over n to the s where n, say, is t squared or something. To be on the safe side, something like t squared. And then this guy without a doubt looks something like the integral du over u to the s. Okay, that's easy. And this one is a mess. And the best we can do really is take absolute values so you get one over n to the sigma, so big O of a sum. So this is a problem. And there are different ways of coping with it. So let me just say when we put the log, this looks fairly benign with sigma greater than one, but now when I put on here a log n to the k, I mean the same arguments apply when I put on log n to the k, log u to the k. But now this term is going to be something somewhat significant. And if you think about sigma very close to one, the sum of log n to the k over n makes this look like a log n to the k plus one. Okay? And the other terms actually are more cooperative, they're smaller. You know, there's terms in the approximation, you've got to figure out all this, but everything else behaves, this doesn't behave. So there are different ways of coping with this. So okay, I guess the first question is, is this a good thing to put into your formula? Well, you're bounding the kth derivative, it's actually log n to the k would be more convenient. It's not actually quite true, but that's the sort of thing you're aiming for. So if we put in log n to the k plus one into our formula, we get something here like log one plus t to the 4k, maybe? Something like that, do you know? 2k, okay. He can get 2k, I can get 4k. So, 2 is better. So you can get something, but it's not as good as the classical zero free region. So in Demetrius' paper, he has the idea that one way to deal with his term is to get rid of it. It's always good. So what's his thought is, that's why last time I said I'm going to regret this notation. I'm going to now try, zeta y is going to be the exact opposite of what it was last time. Oh, I put brackets around it last time, so now I'm not putting a bracket around it. There is a significant set. So n greater than or equal to one, p divides n implies p is greater than y. And here, we don't need to take y very big, it's just like a power of log. Is that right? No, it's going to be like t, because t squared, so it's like n over there. So, we're going to work instead of zeta, we're going to work with zeta, remove the small primes. I guess in Sounds Lecture, he talked about the pain of the small primes. And similarly, one has the pain of the small primes here. But you know, the small primes, so t, in our last example, we could take it as something like e to the root log x. In fact, that's what we chose in the end. So, oh, there it is. Yeah. So, those aren't very big primes that we're removing. If we're counting the primes up to x, removing the primes up to e to the root log x, who cares? So, if we go back to Perron's formula and everything else, and we've just deleted the small primes, it's not going to really affect very much. It's just going to be error term. But here, it means that this guy's gone. Now, the problem is that once you do that, this is not so simple, because you're not summing over all integers n, you got to sieve by the primes less than or equal to n. And so, you've got to compute some sieve estimates, but it's not a big deal. You can do it. Fairly standard techniques. And the matrix ends up with this estimate. It's basically the proof. If you could do better estimates for the case derivative of zeta, then you could do better. So, let me just formulate a version, which I know won't please Jean, but this is what I'm going to write down. So, if you're trying to understand the sum of this sort of sum, as I said, it's a sigma plus IT. And the thing that's interesting, if you like, from partial summation, the log n to the k over n to the sigma, at least when it ends with reasonable size, is not important. It's the one over n to the IT that's causing the cancellation that's of interest. And the estimate from Vinogradov, which turns out to be, is used in Vinogradov-Korobov, but it also can be used here, if you like, just as a black box, is that the sum of n to the IT n up to x, the mean value, if you like, is less than, less than exponential minus log x cubed over log t squared. And this allows you to, well, you can imagine, use partial summation with this, go through the estimates. This allows you to put a two thirds in here. Okay, maybe there's some log-log floating around, but you can put a two thirds in here. Well, you can see the two thirds rearing up there from the log x cubed over log t squared. When you put the two thirds in here and you go through the calculation over here, let's see, what do we have? We're going to end up with a two thirds here, a two thirds here, and then equating log t with log x over log t to the two thirds. So this will be to the five thirds. And so you get log x to the three fifths. So once you've done this, assume this n to the IT estimate as a black box, you can just go ahead and get the strong form of the prime number theorem. And if you do it a little more precisely, you get a log log x to the one fifth in the denominator. So you get exactly vinaigrette of Karabov. So obviously the meat lies in this estimate for proving the strongest form of the prime number theorem. This is not something easy to prove, but it doesn't use analytic continuations, a zeta function to the left of one. It does achieve our goal of getting the strongest form of the prime number theorem without venturing to the left of one. Excuse me? How do you know about the zeta function? Oh, well, okay, one over log. Let me not get into that. So, I mean, it's the first step, though you need a little more. You can get your lower bound on zeta simply by zeta sigma cubed, zeta sigma plus IT to the four, zeta sigma, again, used Maten's identity. And so since we have upper bounds on zeta to the right of one, we have a lower bound on zeta as a consequence. And I think this is quite sharp enough. So you've got to, if you start doing it with removing the small prime factors, it all works out very nicely. Okay. So let me go back to the distance function. And yeah, so, I mean, in some sense, it's a little disappointing. So Demetrius found this brilliant argument. It gives you, if you like, a pretentious proof of not only the prime number theorem, but the strongest known form of the prime number theorem, lots of innovation, lots of clever ideas, and you end up with the same result. I mean, it's such a weird result, log x, the three-fifths, log log x, the one-fifth. Well, the reason why is obvious because the deep idea that's needed to get there is the same deep idea. It sort of tells you something about, I mean, I haven't really said this before, but let me now be a little negative about what the history of working on prime number theorem from the perspective of Riemann's zeta function is. What is it that we're really doing when we prove zero free regions in the classical way? So you could argue that, and especially if you look at the proofs, you could certainly argue that here's the one line. Here's the zero free region for the Riemann's zeta function that we know. Here's the half line over here. Demetrius' proof gives the same zero free region as I pointed out. It's easy to do from this proof. Here we've sort of avoided going across the one line, although we can note that you get the same zero free region. But what are the classical proofs is, in a sense, what they do is they say we know the zeta function so well to the right of one that in order to know it just to the left of one, we can surely extrapolate. And we have to really clever to get as far as we can go. That's roughly what happens. So sadly, most stuff about zero free regions, and I hope I don't get shouted at by Henrik, is about extrapolation to what happens to the right of one. So there is some logic to never going to the left of one, because most of the consequences are more elementary statements. Okay, I guess though at the end of a day, this new approach, if you like, which isn't so new, but writing it as one whole thing is perhaps new. And the classical approaches are really equivalent in many, many ways. But there are some advantages to the new approach, which I'm going to try and point out. So let's go back to defining the distance function. I'll do it over here. This doesn't look like mine. So typically, the distance function I'm going to be interested in. I gave you a more general definition.  They live inside the unit circle, and we'll define the distance between f and g with respect to a limit x is, the square of that is the sum over the primes up to x of one minus the real part of f gp over p. So in the proof of the prime number, oh, maybe I'll go a slightly different direction. So in what we talked about yesterday is, part today, is that the meter, the prime number theorem lies in integrating zeta prime s over zeta s. And understanding that as we, well, times x to the s over s and understanding that. And as we just saw, working with zeta prime s is not that difficult. I mean, it's a pretty well understood function. It's just the sum of log n over n to the s. It's not something to fear, especially to the right of one. So the element of fear here is not zeta prime s. It's not x to the s. It's not s. It's the one over zeta s. So you can actually show that getting good estimates to that is equivalent to getting good estimates to one over zeta s. There are elementary ways to do this, but let me, let me not get into that. One over zeta s x to the s over s ds. And then if we use Perron's formula again, well, for one over two i pi, which I've missed, but you end up with the sum of the Mobius function up to x. So in fact, the prime number theorem is equivalent to, it's the same statement as that the sum of the Mobius function is little o x. So square free integers are as likely to have an even number of prime factors as an odd number of prime factors. And in fact, the bound on mu of n you get is exactly the bound we've been doing with the prime number theorem. I mean, at all, analytically, much the same arguments. Now, unlike mu of n, because mu of n is the multiplicative function, and that leads us into a whole different field of things we can work with. And so a question one might ask is not just when is mu, the mean value of mu, or the sum, and one way to interpret that is the mean value of mu of n is little o one. It tends to zero. But you can say, you can ask that in a larger context, which is when is the mean value of any multiplicative function going to zero? So can you name your multiplicative function whose mean value is not zero that lives inside the unit circle? One, enter the it. Thank you, Heidi. Okay, so one is a good one. That doesn't have mean value zero, but enter the it is a great one. Okay, forget everything I said before about the difficult bit at the start because it's not relevant. This is more or less, oh, we want the mean value, so one over x. This is more over at less one over x times the integral of u to the it dt between zero and x. And this is this is just equal to one over x times x to the one plus it over one plus it. And then when I divide by x, you get this. So for those of you who are taking notes, I'll give you a second to stare at that because it's something worth thinking through. Thank you. It's also worth getting correct. I didn't heckle you very much. Okay, so this is kind of a cool mean value because look, if I take the absolute value of x, the it over one plus it, it is what one square root of one plus t squared. It's some number, right? So when I take the mean value of n to the it, it doesn't tend to zero because if I take the absolute value, it tends to some constant bigger than zero. But what's beautiful here is it doesn't tend to anything, the mean value, right? X i t as x grows just goes around and around the circle. So we're in a little bit of a strange place with mean values, multiplicative functions in the unit circle because they don't necessarily even tend to a limit. Anyway, this is another example, class of examples. This is for any real t that the mean value does not go to zero. Can anybody give me any other example? The answer is yes. So what if I took one, so one is obviously a class from that class, take t equals zero, but instead I took f of n to be, f of p to be one on every prime except 17.  So when I take the mean value of this thing, it's going to be some constant. I don't know what the constant is, but it's clearly not going to be zero. It's going to be something, all the terms are going to be of a decent size, or almost all the terms. Mew squared. Yeah, another one, nice one. So it's not hard to find other examples where the mean value does not go to zero, but the mean value, but those examples are, take the nice, pretty examples and mess with them. That's what it boils down to. So what is the limit of how much you can mess with the pretty examples of n to the i t? And this is what, well, we'll see the distance coming in in a minute, but you can ask, are there any other examples other than the n to the i t's and the mess of them a little bit, but doesn't really change, it just changes the constant. Are there any other examples? And the great theorem of Hollass is no, there are no other examples. So Hollass, this isn't going to obey, is it? Will you stay there? It's exciting. So Hollass's theorem says, if Hollass, I guess it's the vibration set it off, Hollass's theorem states that if, so f n multiplicative, f n less than or equal to 1, if the mean value of f of n does not tend to zero, then f is pretty close to being an n to the i t. And what does pretty close mean? Then there exists t real, actually t bounded even, such that the distance between f and n to the i t, going all the way out to infinity, is bounded.  So this is what we mean pretentiousness. For f pretends to be n to the i t, I mean maybe it is n to the i t, maybe it's close to, you know, we could have used the word is approximately, but that's used so often. So pretentiousness. There was something that was more surprising when we initially coined the phrase of two things, really had no right to be the same. Okay, so that is what happens for general multiplicative functions. And the question in the prime number theorem is for mu of n. So we said that the prime number theorem is p n t false would be the same thing as the sum of mu n does not go to zero, which by Holast's theorem implies that mu pretends to be n to the i t, which as we said last time, from the triangle inequality, this tells us that we can just take the squares here. So mu squared is 1, n to the 2 i t pretends to be 1. And that would imply, for instance, well, there's many ways to prove that's nonsense. But one way to prove it is just to prove that would say, data has a pole at 1 plus 2 i t. So Holast's theorem, if you like, fits in to the proof of the prime number theorem very nicely, the classical proof. But here's the great thing. When you're dealing with the classical proof of the prime number theorem, you're using zeta and its properties, its analytic continuation, that it's analytic everywhere, that there's a real structure going on there. With Holast, you don't have to worry about structure. Just any multiplicative function, the same notion of pretentiousness, is all that's important. So it allows you, in the same class of kind of questions, to generalize vastly from coefficients of a Dirichlet series that has Taylor series, essentially, to Dirichlet series that don't necessarily have to be well behaved. And that ends up being an enormous difference in what you can hope to prove. So let me say a little bit more about this distance function. So let's have a look at this Dirichlet series. So let's suppose that f of n, you know, I said last time that when we do a problem where we're counting things up to x, we're not really concerned about the value of the function at primes beyond x. So when we're looking at the sum of a n and up to x, we can play with the values of a n for n greater than x. So let me assume that f of p to the k is 0 if p is greater than x. And what I want to look at is f of s, so the sum of f n over n to the s. So now it's a multiplicative function, not like here where we were taking an arbitrary Dirichlet series. And we're looking at the product of 1 plus f p over p to the s plus f p squared over p to the 2s, et cetera. But this is only over the primes up to x. So that gives us something nice and finite to play with. And where we do our integration, where we're working, we're having the real s is c plus i t and c is greater than 1. So if I want to understand the size of this, just like in Sam's lecture, the terms after the first one are all like 1 over p squared. They converge. So up to a constant, they're not relevant to the size. So up to a constant, this looks like exponential of the sum p less than or equal to x of f p over p to the s. And I've written this as p to the c plus i t. So it looks like that. So that's really just f p over p to the s rewritten. And now remember c is awfully close to 1, so I'm going to cheat a bit and say it's 1, but actually it's a cheat that's allowed because it's 1 plus 1 over log x. Again, it changes things only by a constant. And now this is a bit like when we did the distance between zeta s and zeta s0. I'm going to add in, if you like, a s0 term. So what I'm going to look at is, actually I'm going to do something weird here, start off with. So I'll put this as a minus and a minus.  So this extra term I've added is a minus sum of 1 over p, which is log log x. I'm going to exponentiate it. It's a log x. So I want to put a log x in there. So this is up to a constant. So whatever. You can see where I'm headed, I hope. And so now let's take absolute values of f of s.  We take the real part, x of the real part of this. Real part of 1 is easy, real part of a second, but it's not so easy. But anyway, this is log x times exponential of 1 minus the real part of f p p to the i t. But that's exactly this. Okay, so this very much tells you that this distance function is got to be relevant to understanding these integrals, this Perron's formula. So when we look at Perron's formula, we're integrating up this line just to the right of 1 of f of s x the s over s. And as we said, our approach here is just to take absolute values in the most moronic way to try and understand. We'll put some bits in to make it a little less moronic, but the key is going to be understanding the size of f as we go up that line. And the size of f, well, here we have a very good guide to it. So what's the maximum the size of f could be, say, up to height t? It's going to be reflected by the minimum that the distance can be. But that's much like what happens in whole s's there, because that's talking about minimum distance, right? There exists an n. There exists a t such that, so this will be the minimum distance, if you like. So actually, I just realized, looking at the audience, that I should say about Cuclops's proof that if you've looked at the book of Van Janssen-Koelsky, since they're both here, I need to say this. And it was certainly the inspiration. If you look at it, there's an elementary proof of the prime number theorem, or sort of a bit like this, sort of elementary. I mean, it uses ideas of a zeta function. And what you'll see, there are many elements of what I talked about in Cuclops's proof. And in fact, the origin of his proof was when he was a postdoc. He's now a professor at the University of Montreal. But when he was a postdoc, we exactly talked about combining Van Janssen-Koelsky with pretentious ideas.  He was more pretentious in here. OK, so this is Hollasser's theorem, and it's central to this subject. And as I promise you, I want to give you Harper's proof of Hollasser's theorem, and or slightly edited by Sound and I. Oh, I'm so lost with these boards. Will I ever get this down?  You have to figure it out. Always look so effortless when you do it. OK, so, yeah, let me be a little more precise then about this theorem. So m f x t will be the maximum with t less than or equal to t of f c plus i t. So the maximum that the Dirichlet series gets in absolute value, where we're considering the integral. And as we said, this is like exponential of minus log x times, sorry, log x times e to the minus the minimum, t less than or equal to t, the distance squared from f to n to the i t, up to a constant. So, OK, so that's the connection. It'll be useful to have that. And now let me, I've given you sort of the vague version of Hollasser's theorem. Let me give you the less vague version of Hollasser's theorem, which is the uniform upper bound on f. Now again, if you look over there, you've got, if you look up there, the sum, take a n as f of n. You've got the Dirichlet series x to the s over s. So the x to the s looks like x. OK, that's expected. The 1 over s is probably benign. So the main thing is the value of a Dirichlet series, the biggest it can get. And Hollasser's theorem confirms this in that he actually proves that the sum n less than or equal to x of f of n is less than, less than this maximum that the Dirichlet series can be times 1 plus logarithm of that. So I'll explain that in a minute. And here we have to take some like t greater than or equal to log x. There are other versions of its theorem, but let me just state it this way. So it's an explicit way of doing the same thing. What I should have said about, something's wrong with this. I'm sorry, I meant to, I'm sorry, my notes not brilliantly written, which isn't the help to you. But let me do, it's natural to divide through by log x here. OK, that's what I should have done. And then this is less than, less than 1, this maximum. So it's a maximum Dirichlet series divided by log x. Log x is the maximum it could be. Well, you can see that because that's e to the minus the square of a real number. So it's bounded this maximum. And then log, I guess I should put absolute value since log is going to be negative here. So if you like, suppose this is 1 over 1,000, because it's less than 1, this is pretty small, it's the 1 over 1,000 you care about. So the focus is on this, that you get the maximum of the Dirichlet series divided by log. There's a little bit extra, which you can concoct an example to get this little bit extra. I probably didn't have, I did plan to do it, but I possibly didn't have time to do it. Maybe I should tell you some nice examples of, of Paul-Assisterum. I guess I only have 10 minutes, so it'll be limited what I'm going to do. So, so perhaps the most, so you've sort of got these natural examples of, of multiplicative functions that have been worked out in literature like mu n and n to the it, but there's one that's particularly lovely. And it generalizes mu n, but in a surprising way. So mu n is the multiplicative function where fp is minus 1 in all the primes. I'm not worrying much about the p-squads because they tend to only affect the constant, they don't really affect the meat of the issue. So a generalization of this would be make fp be alpha on all of the primes. Okay, so nice generalization. And if you do that, well, what Dirichlet series do you want to work with? So I said that working, that what happens on the prime squared, prime cubed, doesn't really affect things very much. So you want to pick, they're easier Dirichlet series as possible. What Dirichlet series is really easy in the sense that we are familiar with it, which is, has every fp equals alpha zeta to the alpha. So there we are. If f of s is zeta s to the alpha, then every fp equals alpha, right? So this is the subject of, I think it was originally worked out by Selberg in his referees report on Satay's paper. And this gives you, which was subsequently published, I should say. So that the mean value of this f of n, where it's alpha on every prime, looks like, and I'm not going to give you every single detail, but I'll give you the important bit. c alpha over gamma of alpha plus little o one times x log x to the alpha minus one. Huh? Oh, okay. Yeah. And c alpha is benign. Benign means well behaved, doesn't worry us. So this, okay, so what? Well, let's just verify it in cases we know. Alpha equals one, okay, that's pretty good, log x to the zero, so one and times some constant. We know the constant's one, that's easy. Alpha equals zero, the multiplicative function that's zero everywhere. This term's only log x to the minus one, that's not really telling us, because we know the mean value of the multiplicative function zero is actually zero. So this has to go to zero, but gamma of zero is a pole of gamma. In fact, what do we know about gamma? Gamma's meromorphic and it has its poles at alpha equals zero minus one, minus two, etc. So this also accounts for the mu, the prime number theorem, right? If we take f is minus one, f of p is minus one. We also get that this is zero and this thing's irrelevant. So you've got a sort of balance here. At the very special points where f of n is zero minus one, actually minus two, that's outside the range I've been talking about, you do get the mean value zero. At every other complex number, it's a non-zero, I mean it's something non-zero times some particular power of log. Let's see how that compares. How big can zeta be? Yeah, so if you compare this, you've got the zeta s to the alpha, okay? And that's going, so then what we want to do is know how big is this function m. So it's going to be zeta s to the alpha. How big can that be? Remember this is zeta s where we truncate it at x. So that can be as big as log x and then to the alpha over log x. So Holast's theorem gives you the log x to the alpha minus one in the upper bound and there's just some log log extra that it gives, which we won't worry about. So Holast's theorem really does well in Selberg's beautiful example. So, well, if you look at our book at some point, you will find that you can concoct an example of log log by playing with the example where alpha equals i and changing f on the big primes. Anyway, I won't get into that here. There is an interesting question which says, when is Holast in the right ballpark? When does it give you an upper bound of the right ballpark? Because we know two examples where it's nowhere close to the right answer. The mean value is much smaller. If f is always zero, the mean value is much smaller, much more interestingly. If f of p is always minus one, the case of the Mobius function, the mean value is always much smaller than the Holast prediction. So if f is in the unit circle, is there some way to predict when this kind of fails up here? And another achievement of Cuclopolis, which I call the Cuclopolis, I'm going to have to be a bit vague because it's somewhat technical, converse theorem. So inside the unit circle, so if you like alpha, obviously f's real part greater than or equal to minus one. So the size of this thing is bigger than log x to the minus two. So what it says is something like the following. If consistently, uniformly, you have something like the mean value is less than, I don't know what the latest is, is it log x to the two plus epsilon or three or two? Two, exactly two? Two plus epsilon. Fantastic. Almost as good as possible. Then, f pretends in a certain sense, it's very similar to that, to be either zero, at least on average, or mu n n to the it. So, yeah, the only way you can sort of get a mean value of multiplicative function to consistently not be close to the right answer in Halas is in these very special cases that we already know about. Okay? So, anyway, let me not go further on that. See if everything I wanted to cover is there. It's ridiculously optimistic. Okay, maybe I'll just finish with talking about the new proof of the triangle inequality for this distance function. So, may come as a great shock to you that Terry Tao emailed me last night with a nicer proof than we had. So, if these guys lived on the unit circle, f and g, then if you look at f minus g, f of p minus g of p squared in absolute value, then what are you going to get?  So, twice one minus real part of f g conjugate. Okay? So, when it's on the unit circle, you can convert this into some sort of regular norm. Okay?  Well, it is this. Okay. Okay. So, that's exactly the same thing as the distance function. So, the norm of f minus g is the same as the distance function. And now, we know we're comfortable with such norms, but we're only playing on the unit circle. And we know with such a norm that you have the triangle inequality, g minus h. And these are just for things on the unit circle. But, of course, with norms in this sort of, you know, Euclidean type norm, you have linearity. So, Terry's nice idea, which nobody had seen before, was that if you do have a multiplicative function with a value in the unit circle at the prime p, then why not treat this as the average of, say, two multiplicative functions that have values such that that's the mean? Okay? So, you do this at each p, f p, g p and h p. You just average. You can call this a probability argument whatever you want. You average over them all. And then you do exactly this. You just add these all together. You average over them. And that ends up giving the triangle inequality for the distance function. Is that reasonably accurate description of your argument? Yeah. Yeah. Okay. Well, I'll finish with Terry's argument. And, where did you succeed? The problem is in 1972 or so. Right. There is a, I mean, I'm not really talking about, there is a rich history of, of this. You can remember. Heidi certainly has some papers involved in that. There's some beautiful articles. I guess, I'm really focusing on where it's taking us right now, which is, you know, it's always what you want something 30 years on as found its niche again. 50 years on. Yeah, I guess 1970s, 1670s. 69 far less. Is there an inversing of 67? 63. Okay. So, on the 50th anniversary of versings there.