You know, we hear a lot of things about Islamic fundamentalism in the media as this, you know, great bogeyman. But what's really interesting is, compared to how much time is spent on talking about it and, you know, bringing it up in every single day, we never really hear where it's coming from and how it's kind of formed.  But that's a really good question, why is that so? Why are we never told where did this phenomenon come from and how did it develop? And the truth is, well, very uncomfortable for the people, the ruling class in the West, because, you know, even if you look at today, for instance, this morning I hear they were going on and on again about a assumed chemical attack in Syria against Idlib, which is the governorate that the Syrian regime, you know, allied with the Russians, are attacking at the moment. And you know, we're told that the people who have power in this area, in Idlib, you know, freedom fighters, rebels, moderate rebels, whatever that means. But what we're never really told is that these people are effectively, that whole region is effectively controlled by a branch of al-Qaeda, which has many, many different names over the time, but which is essentially a branch of al-Qaeda. In, for many years, in the beginning of the Syrian civil war, we were always told that what we have here is a, is a, on the one side we have the brouh al-Assad regime, we agree that's a very brutal regime, and on the other side we have these freedom fighters. And of course this was true to begin with, but at a certain stage this movement became, became sectarianized and was hijacked by Islamic fundamentalists, by the same people who, you know, later suddenly became known to everyone as ISIS and who were suddenly the worst bogeymen in the world. They were there all alone, supported by the West, supported by the allies of the West in the regions, by Saudi Arabia, by Turkey, by Jordan, and the CIA, who had an operation in the country. And even, you know, if we go to Afghanistan today, there are negotiations, we never really, you know, they don't talk about it in the US, but at the moment US imperialism is in negotiations with the Taliban to secure a withdrawal from Afghanistan, essentially handing power, you know, sorting out some kind of power sharing deal with these, these extremists. I would say that more or less at all stages, Islamic fundamentalism, as we know it today, has been tied to imperialism at all stages of this development. From the first days of the modern Middle East, the main form of counterrevolution has been one form or another, in one way or another connected with Islamic fundamentalism. That doesn't go to say that all Muslims are Islamic fundamentalists, but that this is a political movement, it's a counterrevolutionary political movement that at all stages has been dropped up by the imperialists and by the most reactionary forces in the region. Even if we go back, you know, just after the collapse of the Ottoman Empire, when Turkey was known as Turkey today, was occupied by Western imperialist powers, you had the Turkish War of Independence, which was essentially a bourgeois revolution, led by Kemal Ataturk, Mustafa Kemal Ataturk, and which was fighting a struggle against the imperialists, and what was the imperialist's last card, what was the last card that they played, was this old sultan, who was basically a British puppet, nothing more than that, who used his formal designation as a caliph to call for an uprising against this movement, trying to reach out, trying to connect with the most backward reactionary layers of Turkish society against the Ataturk movement, which wasn't leftist in any way, but it was revolutionary and it was a complete break with the backwardness of the past, trying to modernize Turkey, trying to bring it into a new era. And we saw that the bourgeois revolution in Turkey, and what we'll see throughout the history of the Middle East, the main task of the revolution can only take place under the banner of secularism, the division of state and religion. And that's more or less just like it happened in Europe, but there's a big difference in that this time strong capitalist powers do exist. So this is not like the Enlightenment period, the rise of the capitalist class in Europe, which was fighting against the Catholic Church. This time there are actually more powerful powers in Europe, which have already developed capitalism, and which do not have an interest in an independent, strong capitalist society developing in the Middle East. And that's the irony of the whole thing, that you have the European bourgeoisie coming to power fighting against obscurantism, against religious fanaticism of the Catholic Church, and in the Middle East they lean on these exact type of forces, and lean on the most conservative layers and classes, even classes going far before, who were established and who belonged to eras which are far before capitalism in order to dominate the region. European capital had no interest in developing an independent, strong capitalist Middle East. Insofar as they did help the development of some sort of capitalist class, this class was completely tied to the old order. It was basically the big landowners, aristocratic, you know, one form or another of aristocratic and tribal people. And these people were completely tied to imperialists. They were completely tied to the old regime, to the old order in society, and they were completely incapable of developing society in any meaningful way, and they were hated by the masses. And at the same time the empire, the Ottoman empire after World War I, was broken up and divided in the most humiliating way, and the Arab nation in particular was divided up in exchange between the big powers like change in a poker game or something like that. And these people who all share a common history, there's not a difference at least before. There's not a difference between people living in what's known as Syria today or Jordan or Palestine or Lebanon. It's the same tribes and peoples who lived across these borders. In fact, if anything, the imperialists made sure to cut these up and make sure that they are divided up as many different groups in each nation in order to keep them weak, in order to be able to divide, play them up against each other in order to rule them. And these people who used to live side by side for centuries were divided up and split up like cattle basically. And this humiliation and this betrayal left a mark on the psyche of the Middle Eastern masses which is still there today. And the oppression and the humiliation of the imperialism was obviously the source of enormous anger. And immediately after World War I, you saw that the general crisis in society, the general turbulence in society, opened the way for a whole series of national liberation movements everywhere. Immediately beginning from after the World War I, especially in Egypt, there was a revolution against the British which was defeated. But everywhere you saw in Iraq, in northern Iraq, in southern Iraq, you saw uprisings, mass movements against imperialism, for national self-determination and liberation. And in general, you saw widespread radicalization. But as I said, the capitalist path was repeatedly blocked basically by the imperialists, by the so-called, by the ones who were, in fact, capitalist in the region as well. And therefore, you saw actually the development of movements which even went further and further and further to the left of capitalism. You know, starting off from just the basic democratic demands, they were more and more radicalized, moving to the left. And here Egypt was a key country, especially because it was the most developed country at the time. And in Egypt you have these new cities developing with layers of industrial workers for the first time, although it was a small working class, but for the first time you had industrial working class, you had the beginning of an urban intelligentsia, urban middle class, intellectuals, and top state officials who were being radicalized in this situation by the desire to develop society basically and being blocked at every single step. And at the same time you had, on the other hand, a pool from Marxist or leftist perspective in the sense that the Russian Revolution where the masses, the workers, the poor and the peasants had taken power in their own hands was an enormous source of inspiration.  Now, in Egypt, imperialism was seen as a massive impediment for the development and modernization of Egypt, and that was the main driving force. In fact, especially amongst middle classes, they wanted to have some sort of national dignity and development and modernization of Egypt. That goes for every single country in the Middle East up until this day. But they couldn't because they were faced with this imperialism. Now, at the same time you had also another layer developing, which is what was called in Egypt the lower FND, I think it's pronounced, which is a layer of newly urbanized village dwellers, often people coming not from really poor background, but from basically middle class, medium landowners, who are not the enormous big landlords who have ties to the imperialists, who have direct ties to the state, but the mediums ones which are being out-computed by the state, by industrialization and who feel the pressure from the rising cities who are coming to dominate the villages, the under pressure from the economic crisis and the pressures of imperialism, and then losing their independence. These are people who had a traditional position in society before, like kind of village heads, people would go to them and ask for advice about all kinds of things, like who to marry and whatnot. People would go to borrow money from them and get advice from them if they had economic problems and so on. They had a certain social stature, which was being undermined by the development of capitalism, by the inroads that imperialism was making, and they couldn't really see any way out. These were the most conservative layers of Egyptian society. If they stayed in the village, they were constantly on the verge of bankruptcy. Imagine every single day thinking, you can lose everything basically, and it wasn't people who didn't own anything. They actually had something and they were losing all of it. At the same time, if they went to the cities as well, they would be ruined because in the cities they didn't have the connections to move up. They might be able to get jobs as a teacher or a lawyer or a doctor, the lowest layers of the middle class, but they didn't have the connections to the big landlords, to the imperialists and so on to move up. Everywhere they went, they felt this hatred and this humiliation, and they longed for stability and for the rehabilitation of their own positions. They were dreaming of a return to the good old days, which is always what the conservative Petit Bourgeoisie does. Being under pressure from all sides, they can't understand the working class and the communists, and obviously they can't deal with the big bourgeoisie. So they kind of dreamed their ways back to this old society where they had a certain position. And it was amongst this layer that the Muslim Brotherhood was formed in Egypt in 1928 by Hassan al-Bana. And the ideology, obviously there's lots of different parts of it, and it is a developed ideology. But basically from our point of view, what it represented, from a class point of view, what it represented was a return to the early days of Islam, kind of a revivalist, puritan view of society. And that became a focal point for this layer within the cities and also the conservative layers which were in the villages as well. Nevertheless, the Muslim Brotherhood was not the dominant force in society. The dominant movement was still to the left. Like you can see, actually the communist party in Egypt made many, many mistakes and didn't ever really grow to become a mass force. But you had all kinds of organizations instinctively, you know, pressured by events, basically moving to the left and taking on a more and more socialist Marxist, leftist rhetoric and ideology. But, as I said, the communist party didn't manage to capitalize on this mass movement to the left, which was especially after the defeat of the 1919 revolution. You saw very, very big radicalization. People saying, you know, this is not enough, we need to break with the whole thing. But as I said, the communists couldn't take advantage of this. And therefore the mass movement on the streets, and there was a mass movement developing, lacked organization, lacked a political focal point. And so this vacuum was filled by a group of officers led by Gamal Abdel Nasser, which was called the Free Officers Movement. And this stage, a coup in July 1952 against the Farouk regime. The Farouk was the king, was basically a puppet king controlled by the British. And the reason why, well, the character of this movement, of this officers movement was, you know, these were people who were deep within the state apparatus. They were high within the military, and therefore they could see all the rot, they could see all the incompetence. In 1948, they lost the war against, you know, all of the Arab nations lost the war against Israel, which was based purely on, you know, on the incompetence of the ruling classes and the deal that they were making with different imperatives. And they could see all of this, and they wanted to rehabilitate their nation, basically. They also had a higher education, they had access to education systems in Europe, and they had a higher cultural level, in essence. And therefore, and also they were organized, as one of the only organized forces in Egypt. And they took power, and initially, Nasser was not a socialist, or even a leftist. In fact, just after taking power, he cracked down on a communist-led, massive textile strike. But he was a nationalist, and Nasser wanted to restore Egypt's honor. That was basically the main driving force of his rule, at least in the early days. But in doing so, so he would increasingly come into conflict with Western imperialism. And he would increasingly find himself leaning on the Soviet Union, which, for him, also was kind of a model which was appealing. Here you had a military-run society, I know democracy, there's one guy from the top that decides. But also a society which had managed to modernize itself from extreme backwardness and barbarism to being the second power on the planet within a very short period of time. And you saw him drifting towards the left, and taking on a left-leaning Arab nationalist, Arab socialist point of view. And he became a focal point throughout the whole of the Middle East for all of the movements developing in the whole of the region and even beyond. Now this movement was radicalized again in 1956 where Nasser, because of conflicts with the British, ended up nationalizing the Suez Canal, which was built by Egyptians, which was run by Egyptians, and the only thing that was not Egyptians was the Prophet of it. And Nasser wanted to use the Prophet from the dam, from the canal to modernize Egypt, to industrialize Egypt. Being blocked by the imperialists, he nationalized it, and then he was attacked by a coalition of Israel, France, and Britain, which he defeated. Again, this further pushed him to the left, radicalized not only the movement, not only the mood in Egypt but throughout the whole of the Middle East against the imperialists and further to the left. Now Nasser actually, he was never Marxist or socialist, he didn't implement socialism, and that was his mistake in fact. If we have to go and study the mistakes of what has led Egypt to where it is today, because Nasser never finished what he started. He nationalized the main industries in Egypt, and by doing so he developed free healthcare, free education, the railways. In fact, you go to Egypt today, nothing has been built since Nasser's time. And that says something about Egypt, about Nasser. But he still maintained the old bourgeoisie, the bourgeoisie remained, and capitalism wasn't in fact overthrown. Nevertheless, he moved far to the left, and in doing so, he became increasingly into conflict with the Muslim Brotherhood. And the Muslim Brotherhood, as I said, represented this conservative religious layers who were not interested, and who were interested in maintaining the power of, you know, had this revival as basic ideology, and had an interest in maintaining the power of these layers, which were being undermined by Nasser's industrialization. His middle class layers, that's the bourgeoisie in the villages, in the cities, and so on. So whereas in the beginning, Nasser actually gave the Muslim Brotherhood positions within the regime, top positions within the regime, like Sayyed Utfour, like the main ideologue of the Muslim Brotherhood, had like top state positions. But as they came into conflict with each other, he started to crack down on it. And that actually radicalized the movement from being kind of a conservative, bright conservative movement, to becoming more and more extreme, to going more and more into terrorism and into like, you know, armed struggle and so on. But nevertheless, Nasser was in no mood to share power, and his extreme popularity meant that him cracking down on the Muslim Brotherhood and essentially destroying them didn't really cause anyone to complain. Or, you know, it wasn't at least not that really good. Yes. Now, Nasser at that point was the most popular man in the whole of the Middle East. But that was because he reflected, he reflected a general movement which was being seen everywhere. And this process actually went further, for instance in Syria, it went even further than Egypt, with a group of officers taking power through a series of coups moving to the left. And actually nationalizing everything, completely expropriating the bourgeoisie, and essentially getting rid of capitalism, as we know it. You had left-wing developments in Iraq, in Libya, in Yemen, there was the same type of movements developing. And you saw the Soviet Union gaining influence as the British were losing out. And the Soviets were stepping in and filling this vacuum. And this terrified the new world power, US imperialism, and obviously also the ruling classes of the region who really terrified where this whole thing was going. The king of Jordan, the kings of Saudi Arabia, the Gulf states, and so on. The Americans, in fact, tried to convince the Iraqis to attack and invade Syria, which was a country, as I said, which had gone the furthest. And in fact, within a few weeks or months, Iraq was in the middle of a revolutionary movement itself. And so, from the military path, the Americans realized that they needed to take the political path. They needed to prepare the political ground for counter-revolution. And the main architect of this was the US Secretary of State, John Foster Dulles, who saw the role of political Islam, who saw the role that religion could play in galvanizing a layer of people, a galvanizing movement to fight against these things. And he found a partner in Saudi Arabia, which I'll explain later, and actually he was petitioning the Saudi king for calling for a jihad against the Syrian regime and so on, but the Saudi king at that point said, we cannot do that. We cannot go against these regimes publicly because the movement was so strong also in the Arabian Peninsula. Now Saudi Arabia is a peculiar nation because in many ways it's an artificial formation. At least it's a formation that wouldn't last as long if it was just left to itself. It's basically based on an alliance of a clan, the Al-Saud clan, and the Wahhabi clergy of the Sheikh Al-Sheikh and the Sheikh clan, I think I called, and the Wahhabis who have this extreme Puritan form of Muslim ideology, they basically made a power sharing deal whereas the king and this Al-Sauds basically controlled the state, they're the king and they controlled the state and the Wahhabis formed the ideological of that state. Now this probably could survive for a very, very short period of time if it hadn't been due to the support of first British imperialism in order to have a foothold in the region and later on the Americans who used the Saudis and the Gulf states on the one hand as a political tool of counter-revolution but also later on as a source of oil as we know it. Now the Saudis were terrified of the movement that was going on, of the development that was going on in the region. They had taken in many of the Muslim Brotherhood people who had been escaping Egypt and they had been helping to maintain the network of the Muslim Brotherhood in the whole of the region. There was no problem in that this activist organizational side of the Muslim Brotherhood they found a common ground with the Wahhabis and this basically strengthened both of them and radicalized the Muslim Brothers but also gave the Wahhabis a more organizational, political kind of character. Now the fundamental problem in Saudi Arabia, in the Saudi regime is that on the one hand you have the ideological arm of the regime calling for a caliphate for an Islamic state which does not have a king and on the other hand you have the same country ruled as a kingdom by a king who also is no secret does not live in any way as to prescribe by Islam and certainly not that of the type preached by the Wahhabis. And therefore these groups increasingly they come into conflict many many times but throughout the 60s and 70s this became worse and worse. In fact there was at a certain time in 79 there was an uprising where a series of Islamists took over the grand mosque of Mecca and led to several hundred dead people but all of this was a warning sign to the Saudi royal family where this whole thing was going that they were actually faced by uprising internally a right wing reactionary uprising against themselves and so they made a deal with the Wahhabis which was basically if you just leave us alone in Saudi Arabia we'll give you tons of money and everything we can helping you set up schools and so on throughout the region and hand in hand with again the aims of US imperialism which was looking for a base of support to counter the left word drift that we saw in the region and the first time that this was tested was in Afghanistan in 1979 and it started in 1978 in fact in the Sauer revolution were similar to Egypt a group of army officers left wing army officers took power in Afghanistan and immediately implemented a whole series of things such as banning the buying and selling of women canceling all debts that the poor peasants had to landowners expropriating 75% of land which was held by the big landlords and other progressive measures such as literacy programs and so on they also made many many mistakes but it was nevertheless a huge break with the past that Afghanistan had endured and the regime immediately fell under the in the sphere of influence of the Soviet Union and the US the way that the Americans countered this was by basically forming an alliance with the Saudis and the Pakistani intelligence again leaning on the most backward layers in Afghan society with its clan leaders the big landlords and you know forming what is called the dollar jihad which is the channeling of billions and billions of dollars into this movement of the Mujahideen which later on became the Taliban as we know it I think the figure is 3.2 billion billion dollars and a lot of this money would come from Saudi Arabia who also would provide this arms and so on and the CIA was kind of coordinating this whole thing the war that lasted more than 10 years and eventually ended up with barbarism in Afghanistan and I don't know if you remember they talk about al-Qaeda and all these people these extremely evil people now but back then Osama bin Laden there's a very famous article interview of Robert Fisk with him funny enough he's like the most anti jihadi person now but it was like a freedom fighter that was the title that they had with the picture of Osama bin Laden next to it and I don't know if any of you guys remember it but Rocky what's it called Rambo Rambo which is an action film from the 90s or 80s it's actually dedicated to the Mujahideen it's dedicated to these people to these animals that was the west pushing it now all of that is of course forgotten anymore and of course once the business was finished the Taliban remained and became a headache for the Americans but nevertheless for them the main point was the class questions was the defense of the property rights of the capitalists and the ruling class in Afghanistan and elsewhere because Afghanistan wasn't that important but it was a model for other people to follow to expropriate the big landlords the banks and the capitalists to run society without any private property and this guy Brzezinski who was one of the main architects of this they asked him recently in an interview what about everything that happened since and he said this what's important to the history of the world the Taliban or the collapse of the Soviet empire some stirred up Muslims or the liberation of central Europe and the end of the Cold War and this shows exactly the way that the Americans and the imperialist thought but they not only destabilized Afghanistan they destabilized Pakistan they also in the same period in 65 they supported in Indonesia the massacre of 1 million communists by mainly Islamist backed groups and the Indonesian regime in Palestine the liberation movement in Palestine against Israel was predominantly a left wing movement and it's documented very much how the Israeli and the western intelligence agencies everything they could hounding down the lefties blocking them every single, every chance they could while allowing Hamas the Muslim Brotherhood to set up mass to preach this to disseminate basically because they knew that the Islamists while they might be a headache they weren't a fundamental threat to the fundamental interests i.e. of capitalism they couldn't in any way appeal to the Israeli working class which is what the Israeli regime is terrified of that the Israeli working class somehow would turn against the state itself if we come up a bit closer to today in Iraq before the Iraq war there wasn't any there wasn't no, you can say what you want about Saddam Hussein and he certainly wasn't a pleasant man but there wasn't al-Qaeda and the form of sectarianism that we see today was nowhere to be seen in Iraq, in fact the main traditions in Iraq were leftist, again communist leftist Arab nationalist traditions and also the Kurds had leftist traditions in Iraq but after the Iraq war in order to dominate, in order to control the country the Americans talked about winning the hearts and minds what did that actually mean in practice that meant stirring up tribalism sectarianism playing out Sunnis and Shias and Kurds who've been living together peacefully for centuries we never had any problem you didn't ask what people were before all of this you might know but it wasn't like a big thing most Iraqis from that generation, from the generation before me they were marrying and mixing across all of these things but the American pushing sectarianism pushing this form of nationalism and tribalism in order to fight down the anti-American uprising and fight that was brewing supported by the Saudis and when the Arab revolution came this is even more recently the Arab revolution also had a big impact in Iraq in huge demonstrations in the east sorry in the west where the Sunnis are based especially big big mass mobilizations taking over squares and they were non sectarian they were for democracy, for democratic rights against sectarianism, against poverty against inflation, against unemployment but what did the Saudis and the Gulf states supported by the Americans do they poured in billions of billions of dollars supporting especially tribal forces but also these sectarian Islamist forces which later on fed into the development of ISIS as we know it and reinvigorated al-Qaeda in Syria we had the same thing first of all obviously the Syrian revolution was a genuine revolution to begin with but because of things that we can't really discuss now it went into an impasse and that allowed the counter revolution to step in again with the scourge of sectarianism the Assad regime opened up the prisons released all the jihadis in order to push them to actually polarize the society on sectarian basis and to prove this is what the Americans and the West are going to do they want to destroy Syria and they supported the jihadis and of course that is exactly what the Americans and the others were trying to do in fact the campaign in Syria was the biggest campaign in the budget of the CIA and the groups that they supported the so-called freedom fighters and moderates as soon as you went in to look who they actually were they were all Islamist of one kind or another and they were being radicalized more and more in this in this sectarian civil war eventually ended up being two main groups which were ISIS and Jihad al-Nusra or al-Qaeda and all the other ones were dominated by these two there was nothing, there was no moderate groups there was too tiny to operate alone they could be switched off like this but they were kept alive especially by al-Qaeda in order to have like a front where they could receive money and what was the main reasons for this on the one hand in the geo-strategical game that these people are playing against each other, the Saudis, the Israelis the Americans against the Iranians but at the same time it was to cut across the Arab revolution and use it as an example throughout the region which they are doing every single day to say this is what happens when you go on to the streets you know some of our comrades in Morocco were saying people on the streets are saying Alhamdulillah we didn't go that far and we didn't do what the Syrians did because we don't want our country to end up like that and that was the main purpose of carrying out that counter-revolution as it happened and in doing so again they were leaning on what not on the urban working classes in fact the working class in Syria was not really involved in the movement as a class but mainly on tribal rural, backward layers and at the same time also a layer of European youth who has been radicalized by the constant racism harassment every single day that goes on in the media which is supported by the unions which is not really supported even by the left wing and the workers party so we're not saying anything to this just buying into it and that was the main kind of bulwark of reaction against the Arab revolution that the imperialists and the local ruling classes could foster they did the same thing in Libya which is now in a complete state of of barbarism in Yemen they say oh yeah we're fighting against the Houthis the Houthis you can say whatever you want they're certainly not progressive people they're extremely barbaric reactionary but they have some sort of support but who is it that they're Saudi and Saudi attack on Yemen is leaning on Islam al-Qaeda in the Arab Peninsula Islam al-Qaeda in Yemen all forms of Islamist types they've been importing Islamist from Sudan and elsewhere because they can't fight their own struggle now I'm going to sum up in a minute but just there's one more factor in this whole thing is the crisis of capitalism is basically the driving force of Islamism and of the imperialist supporting this because they cannot solve the basic daily needs of the Arab masses on the basis of capitalist society therefore they cannot afford the smallest concessions and they're forced to fight against it but there's another there's another element as well in the Middle East which is Saudi Arabia which is in an existential crisis as I said before this is a civil nation no one I mean people would say I'm Saudis but it's not like the whole country is named after one family no one feels this is my nation I'm going to fight to the death the Saudi army has never been in war it's a puppet army because people wouldn't necessarily die for crown prince or king who lives this extremely disgusting lavish lifestyle whereas the vast majority of Saudis in fact Saudi Arabia is not like the ghost states where everyone are rich there's a huge number of poor people on the one hand 20% of the population are extremely oppressed extremely poor hate the regime then there's the youth and everything living and progressive which is against this dictatorship the authoritarianism the hypocrisy in society women are also against the regime then you have obviously the poor but not getting anything out of this society then you have the Wahhabis and the extremists who also hate the monarchy just from a right wing point of view and so there's no one actually in defense of this country and this country this nation could survive as long as it was supported by the imperialists and as long as oil prices were high as long as the economy was booming it could pay off it would basically pay itself out of social tensions but as the crisis of capitalism oil prices fall you begin to see for instance Saudi Arabia had the first budget deficit what does that mean that means it cannot buy off its network of patronage cannot buy off the jihadis cannot buy off the working class and also it's not as important from the point of view of imperialism the Americans are now self-sufficient in oil Saudi oil is still important but the Americans don't need Saudi Arabia like they did in the 80s where they got 90% of their oil from the Gulf and the way that the Saudis have been acting in the Middle East for instance has also at times brought them into clashing with US imperialism itself for instance in Syria where the Americans at one point said okay we shouldn't support these groups anymore because now they're just spinning out of control and the Saudis kept doing it basically provoking the Americans so in that sense the basis for Saudi Arabia is gradually being eroded and there's no one who's actually going to stand up and defend it which is why it's lashing out which is why the kingdom is lashing out sending out these troops sorry sending out these groups and supporting counter-revolution in one country after another basically trying to keep all these plates spinning in order to divert attention from its own prices but sooner or later all of this is going to come back obviously and I think the days of Saudi Arabia are numbered one way or another it's not going to be what it is today so just to sum up Islamic fundamentalism as we know it today is nothing but pure counter-revolution it has nothing to do with ordinary Muslims in the Middle East who in fact if you want anything the traditional political trend that they would have is leftist, is socialist is communist in the 60's and 70's and 80's you had millions of communists in Iran in Iraq, in Sudan each of these countries you had millions not just as a total in Egypt obviously there was the Arab nationalist trend the same was in Syria, in Libya in Algeria Tunisia, all of these countries the main trend of the masses was a left wing trend and the Islamists were the pure counter-revolution who could never actually achieve mass space on their own without the support of the imperatives it's a reflection of the blind rage and impotence of the Pency Bourgeoisie the tribal layers, the most primitive and also the lumpins, the ones who have been completely dispossessed by the wars and havoc and destruction that it has of course they're downtrodden by the imperatives but they're not finding a lead in the working class and that's the main thing that they're not finding a lead a socialist lead from the working class as it is independently they're extremely weak they find somehow common cause with the imperatives extremely hypocritical because they always go on about how anti-imperial they are but in the end they always end up on the same side fighting the struggles of the imperatives like you know the what do you call it, mercenaries and what it really shows is a complete dead end of capitalism as I said capitalism in Europe came to power fighting against obscurantism fighting against religious fanaticism but in the Middle East it's the exact opposite it's actually leaning some of these tribes this is something that belongs to periods way way way before capitalist developments and for all the radical talk the Islamists don't touch capitalism at all or even fight as I said the role is to cut across the class contradictions in society and they can only gain when the working class is retreating when the Egyptian revolution was moving forward the slogan was we're all Egyptians Christians were forming a ring around the Muslims who were praying that's class solidarity in Syria it was the same thing the main slogan when the revolution was moving forward was we're all Syrians but now that's changed as the revolution met one obstacle after another because of its own mistakes because it wasn't able to take power and it was forced to retreat in that vacuum you saw the development of these reactionary groups tapping into the most backward layers and using them basically as cannon fodder for their own ends so the only way to approach this thing is on a class basis in my opinion the way to fight against Islamic fundamentalism the fight against Islamic fundamentalism and the fight against imperialism and the fight against capitalism cannot be separated the fight for the liberation of the Middle East for democracy cannot be separated from this fight and from the struggle for socialism now that is something that the Middle East and masses will have to learn themselves through their painful struggles but I'm confident that just like we saw this enormous mass movements in the post-war period moving sharply to the left those traditions are and will be revived and reignited and the masses will understand that the only way to overcome the barbarism and the backwardness that prevails in the region is to take power in their own hands and to overthrow the capitalist system


So I want to talk to you about the pretentious approach to analytic number theory, which Sandaraj and I have been developing over the last five years. And some of you were at the summer school. We had some years ago and have access to the book that never appeared on the subject. So for those of you who don't know much about it, let me explain what this is all about. So the general approach to analytic number theory stems back to Riemann's memoir from 1859 and involves studying the distribution of zeros of the Riemann's data function. And as we know, it's been a very profitable approach to L functions and naturally arising questions and arithmetic for a long time. And yet many theorems have been proved without zeros of L functions. And to a certain extent, the community has said, well, those approaches are ad hoc. And look down upon a little bit. Particularly, if you like, the elementary work championed by Paul Erdos. And many of the great theorems of analytic number theory, especially about the distribution of prime numbers, come from so-called ad hoc approaches. So a few years ago, Sandaraj and I had been using certain techniques that we called pretentious. And I'll try and explain what pretentious means in these lectures to prove theorems on open questions to try and develop the subject. And we started to become interested, perhaps, in what established theorems could be proved by these pretentious methods. And we attended a lecture by Henrik Evaniec in Princeton five years ago, where he announced a proof by himself and Friedlander of Linux theorem that in every arithmetic progression where there can be primes, there's a relatively small prime. And Evaniec's proof avoided many of the difficulties about zeroes of L functions that all the previous proofs had. Or Evaniec and Friedland's proof avoided these deep ideas about zeroes of L functions that had previously been part of a very difficult proofs of Linux theorem. Though the proof of Friedlander of Evaniec is not easy itself, but it technically avoids these tremendous difficulties. And if you like, it's an elementary proof, but it will be at a very hard elementary proof. So it gave us courage to say, well, maybe these techniques we've been working on could prove everything in classical analytic number theory. So we sent ourselves that goal, and the first goal was maybe we should try and prove Linux theorem ourselves. And we did. We surprised ourselves that we had some ideas for a technique, and they worked. And we gave a pretty short proof of Linux theorem, certainly much easier than what existed at that time in the literature. And we wrote up, if you like, what we could do in proving the classical results of what you might find, particularly in Davenport and then in Bombieri's large sieve book. And we were able to accomplish quite a bit, but there were some clear flaws in what we had. Most importantly was that we couldn't get a good error term in the prime number theorem. So I'll remind you of what we know about the prime number theorem in the moment, but the number of primes up to x being something like x over log x. And we could achieve an error term that was little o of x over log x, and x over log x is some power, but we couldn't even get that power to be 2. So we were unable to prove that number primes up to x was x over log x plus bigger of x over log squared x. Our methods failed us on that. So that was one big flaw in what we were doing. And that flaw has a knock-on effect, because if you try and prove a Bombieri-Vinogrado theorem, which is central to many of the developments in analytic number theory, then you need to be able to win in the prime number theorem and the prime number theorem for arithmetic progressions by an arbitrary power of log x. So an error term of log x bigger of x over log x over the a is what you need. The second thing was that the subject, and I'll explain all this, centers around what's called Hollass's theorem. And the proof of Hollass's theorem, or Hollass's proof, is a little opaque. It's a little hard to really understand the motivation behind it. And after Hollass, there have been many improvements to his results. In fact, we had a paper improving his results, but all of them started with the same fundamental construction that was somewhat opaque. And this meant a lot of the proofs that we have, well, they worked, but they're rather difficult to understand why they worked in a larger setting. Now, these I would regard as what were the two main flaws in the book we wrote up three years ago. And what's happened in the meantime is that those flaws have both been removed. And not by us, but one by Dmitri Kukulopoulos is here, who showed us how to prove the prime number theorem, not only a strong version prime number theorem, but as strong as can be done by classical methods. And more recently, Adam Harper came up with the new proof of Hollass's theorem that it's not incredibly easy, but it allows one to appreciate what's going on and makes it much easier to use the techniques. So in both cases, we've tried to further develop a theory, developed what Kukulopoulos did and Harper did. And so that's what I'm going to explain to you here. These developments that now means that we feel we have a strong theory that rivals the classical theory. And well, maybe I should embarrass Sandarajan by saying, we'll get the book out in nine months. But no, we've got to work on it and get it out. There are people here I know who would love to see it out so that they can use the ideas. So it will take some writing, because there's a lot of things they're doing. And then people like Maxine keep on proving consequences that one would like to use. So it makes it difficult to write a book when people prove nice theorems to get it finished, the book. Anyway, so let me change the title, because both of these new proofs come down to sort of doing something a little bit surprising. And I'll call this Perron's formula revisited. So that's really going to be my title for the three talks. And in these talks, what I hope to do, if I have time, is prove to you or show you how the proof of the prime number theorem goes the strongest form than a growth of Karabav form, give you the new proof of Hollaster's theorem and explain what Hollaster's theorem is, and hopefully prove Linux's theorem in a way that hopefully you can understand. So it won't use anything very deep. OK, so what I want to do is start with the classical proof of the prime number theorem. And I'm going to assume that you know it already. Now, when one says one knows the proof of the prime number theorem, there's very few people who could actually just write it out, because although the ideas are very elegant, they really are, you have to go through a lot of steps, and you have to make a lot of estimates work in order to make the proof of the prime number theorem work. So I'm just going to quickly go through it, maybe in the next 20 minutes, the proof. And some of the notions that go into the usual proof of the prime number theorem will come into what we're doing. So let's just start off with basic definitions. So lambda n, it's just a convenient weight that we, what we want to do is count primes. And so we want to weight 0 if n is not a prime. And in fact, it's convenient to count prime powers for technical reasons. And if n is a prime power, we count p. This is van Mangels function, as it's called, and it fits lots of formulas. So do the prime number theorem, instead of counting the primes, we count them with this weight. And instead of x over log x expected with some error term, or really the integral dt over log t, we expect the main term of x. And then the question is what error term? And so let me just give 2, maybe 3. So we get big O of x to the half log squared x if we assume the Riemann hypothesis. So we'll get into the Riemann hypothesis a little later today. And the best result known is due to Karabov and Vinogradov. And it's log x to the 3 fifths over log log x to the 1 fifth. So this is unconditional. And that's the best result known, up to the constants. And then the one you've seen in the textbooks, if you've worked through the proof, would be something like this. So we'll more or less give all the details of the proof of that today. And we'll indicate how you get this. OK. So you can get this, but what does he know about? I didn't say that. OK. So let's just remember Perron's formula. So this is a beautiful idea for recognizing an inequality amongst integers. And very simply, so this is the real part of c is greater than 0. And something happens in between. Anyway, the important thing is that, well, it's the average in between. The important thing is that we have a way of recognizing inequalities for positive real numbers using this integral. And hopefully, you've played with it before. So you prove it just in the standard way of pulling the contour to the right or left. And you either get the pole at s equals 0, which leads you to this. And the value on the contour, the added contours, is small. Or you pull it to the right, in this case, and there's no pole. And the value on the added contours is small. So it's not very useful to have an integral going all the way to infinity. So what one wants to do is truncate this at some height. And then one can work out an error term for that, which looks like y to the c. This is all. You find this, for instance, in Davenport's book. And this is if y is not equal to 1, and then if y equals 1, you get some 1 over t. I guess c over t, yeah. So the point is that, well, you can work out the point at which you can truncate and not have to worry about the error term. I'm not going to spend much time thinking about that. You're just going to have to trust me. There is a point. But again, I'm sure you've played with the proof of the prime number theorem, and though that's the case. So what is the key application of Perron's formula? Thanks to Riemann, I guess. If we wish to look at the sum of an and up to x, and let me just assume, we have always have this problem with y equals 1 that it's a bit weird. So let me just assume x is not an integer. So I can be correct and not have to write extra. Then the way we try and sum a sequence of interesting numbers, an up to x, is we write it as the sum over all integers times 1 if n is less than or equal to x and 0 if n is greater than x. But I'm going to rewrite that as x over n is greater than 1 or x over n is less than 1. So the idea is then you can just plug in what we've got up there. We take y equals x over n. And when we plug in y equals x over n, we've got an infinite sum and an infinite integral if I do the thing at the top. So the point having a little flexibility with the c is we can pull it far enough to the right that we can make things absolutely convergent and so swap the order of the integral and the summation. And so we end up with 1 over 2 i pi. Well, let's have a look what's inside the integral. So I've got an and I plug in for here y equals x over n. So it's x to the s over n to the s. So the part that's interesting that varies with n is this. And the rest is x to the s over s to the s. And so what we can replace, we'll just call this thing a of s. And so this is the application of this formula that is most useful in number theory. So in order to understand the sum of something arithmetic, say up to x, all we need to do is understand the integral of a Dirichlet series times something on an infinite line. So you could argue we've taken a nice finite discrete combinator problem made into a hard analysis problem. But sometimes that's useful. OK, so the most famous application will be for the we want to work for lambda n's. And all you have to do to the if you go real part of s greater than 1, then you write out zeta s as the sum of 1 over n to the s. You write it as the Euler product over the primes. Then when you're over in the domain of absolute convergence, you can do things like logarithmic derivatives. And this is what comes out when you do the calculation carefully. So now we can just apply the formula over there. And we find that the sum n up to x of lambda n equals 1 over 2 i pi the integral. Well, let me just curtail it in all in one go. Just believe me that you can for some suitable t minus zeta prime s over zeta s, x to the s over s to the s. And actually, I'll just say what the right error term is, something like this, when you do all that. So again, this is all standard. Oh, there should be a plus log x, because I was careless what happens when you're at the integer. OK, so what's the game now is we have the integral of a rather hairy function. And we have it to the right of 1, because we're doing our manipulations where you can do manipulations without too much worry, where everything's absolutely convergent. And again, we want to play complex analysis. So what we're going to do to this is we're going to pull it to the left. And the hope as we got this integral up to height t is, well, we'll create an integral like this. And the hope is that these integrans are nice and small, the value of the integral and those integrans. And then everything will be picked up by Cauchy's theorem, which is the residues of the poles of the integrand inside here. So we'll talk a little bit about whether or not it's easy to prove that the integral and these lines are small, but in a minute. Let's just think about the poles of the integrand. So what are the poles of this? Well, the room's data function, I'm not going to get into this, but as I hope you know, it's maramorphic. It's got a pole of order 1 and s equals 1. It's the only pole. So at s equals 1, zeta s looks like 1 over s minus 1 plus gamma plus that when s tends towards 1. That's the expansion for zeta. And so that's the only pole. Now, OK, why am I saying that? Well, let's have a look. What are the poles of this thing? Well, actually, s has no poles. 1 over s, there's a pole at s equals 0, but that's pretty benign. We'll think about that in a minute. zeta prime over zeta, we're either going to have a pole of zeta prime or a 0 of zeta. The poles of zeta prime are the same as the poles of zeta because zeta was maramorphic. And then the 0's is zeta. Well, the 0's is zeta. So what do we have? We have the pole at s equals 1. And so the pole at s equals 1 gives us what? Well, if there's a pole of order 1 at s equals 1, zeta prime over zeta has a pole of order 1 at s equals 1, residue minus 1. So that minus 1 times minus 1 is 1. Plug in s equals 1 to here, we get x over 1. So that gives us a residue, x. At s equals 0, we have the residue minus zeta prime 0 over zeta 0. And otherwise, we've got s equals rho, where zeta of rho equals 0. And those are the possible poles. So these two we can handle easily. This requires more thought. So this is the difficulty and the main difficulty and Riemann's approach is understanding these guys. That's what's difficult. And it's still pretty mysterious how to understand these guys. So what happens if you have a pole of order m? Here, when we take s prime over s, we get m over s minus rho. And so minus m over s minus rho. Typically, when we write the formula, we act as if we've got m separate 0's, which makes it beautiful to write that the formula becomes x minus the sum over these rows of x to the rho over rho, counting m times, if it's a 0 of order m, minus zeta prime 0 over zeta 0. You can actually prove this exact formula if you let t go to infinity. But what's more convenient for us is not letting t go to infinity, and then we get the same error term. OK, so this is all very standard. If you know this well, sorry, but I thought it was a school. So it's worth going over to everybody. I'm trying to attribute 0's in that sum. Yeah. So there are lots of 0's, perhaps, to the right of 1. And there's the bunch we all talk about in the critical strip. But there are some along the real axis going back here also. So maybe I have to be a little bit careful, depending on how far back I take the contour. But it's really going to affect things. OK, so now the question is, can we make this argument work? Now, you have to think things through. Like, for instance, when you try and pull the contour across here, what if you happen to bump into a 0 of the Riemann zeta function as you go? Then you're going to hit a pole of this, so it's not going to be so easy to just bound the integrant. So what's a good idea, it seems to me, is if there are 2 0's of the Riemann zeta function, is you take your contour so you sneak your way halfway in between them. Now, is that easy to do? Yeah, it's easy to do because what you can prove, it takes some work, is that the number of 0's up to height t is some constant t over log t, which means there's kind of an average gap between 0's of 1 over log t or 2 pi over log t, in fact.  So when you do that, you can get some idea of the size of zeta prime over zeta. And well, to cut a long story short, in this kind of region, so the real part of s between minus 1 and 2, you can have something like minus zeta prime s over zeta s, is dominated by, well, it's not perhaps not surprising, zeta prime over zeta should be something like the sum of 1 over s minus rho. I mean, that's what appears in the formula over there. And so this is dominated by the s minus rows that are close in. So in other words, the at the rows, this is sum over the 0's rho, such that s minus rho is less than 1. And you can prove plus some sort of error term. So with this sort of estimate, you can work with zeta prime over zeta. Let me not get into that too much. So what I want to do is just focus on the formula here. And see what we can deduce from it. So the sum, of course, I've said psi x over there. So we want to bound the error, psi x minus x. And the main thing here is the sum of x to the rho over rho. And what would be nice is to be clever enough to find some cancellation between the terms for different 0's. But unfortunately, we don't really know how to do that. So the best we can sort of do is something like the sum over these rows of the absolute value of x to the rho over the absolute value of rho. Plus, well, that's a constant. The zeta prime 0 is 0. So we just need to worry about this guy. And I guess I'll take t in a range just so that I don't have to simplify things as we go along. Now, the reason I write this down is simply just to note that the absolute value of x to the rho is x to the real part of rho. Because if rho is sigma plus i t, you can forget the x to the i t. It has absolute value 1. Using the fact that the number of 0's up to height t is like t log t. And also, one can show there are no 0's of the Riemann zeta function very close in to the axis. One can show that the sum of these 0's up to height t is less than log squared, which is partial summation. And so what we end up with here is a bound on this term, which is like x to the maximum part of real rho in this range, rho such that the imaginary part of rho is less than or equal to t, times log squared t plus the second term. So this is the key calculation in getting the error term in the prime number theorem. And well, you can see very clearly that what's really important is how big do the 0's get in this box, how big the real parts of 0's get. Now, what would be convenient is that all the 0's, which is, we certainly know there are 0's on the half line. In fact, billions of them have now been calculated to be on the half line.  If all rho satisfy real part of rho equals a half, or less than or equal to a half because there are some 0's further to the left, then, well, here we've got x to the half times log squared. OK, if we pick t as root x, and this will also be x to the half times log squared, so then take t equals root x. And we get an error term, well, what we said over here, x to the half times log squared x on the Riemann hypothesis. So under the assumption that all of the 0's lie on the half line or to the left. So what we're going to look at in a bit is what happens with other 0's for regions. Maybe I'll just do it rather than leave us around. So what we'll eventually prove is something like the real part of rho in this box. If the imaginary part of rho is less than or equal to t, then the real part of rho is less than 1 minus some constant over log t. And if we can do that and we just plug it in, let's just forget the logs. So these are the main terms up to the logs. And then if we want to minimize this rather, the way to do it is to equate the x to c over log t with the x over t, that'll require us to take something like t as exponential of root log x. And then when we plug that in, we'll get the standard error term on gets in the textbooks. And then this error term requires a better 0 free region, so going somewhat further than c over log t. OK, so what matters to proving the prime number theorem is a 0 free region. And let's just talk a little bit about how one goes about this. So the first step, if you like, is to show that you can't get 0s on the one line and then to come in a little bit. So the standard proof, if you like, which I want to say this. So let's suppose zeta 1 plus i t was 0.  And what is this kind of telling us? You've got the 1 over p weight, and you've got this p to the i t. And arguably, it's telling us something about the way the p to the i t's tend to point, if this was to be the case. So if I was to take, say, absolute value of, no, how do I want to do this? So I guess what I want to say is that if the p to the i t's predominantly pointed towards 1 or the positive real direction, then you might expect this to look something like 1 plus 1 over 1 minus 1 over p to the minus 1. So it would go to infinity. So there's a way in which one thinks that p to the i t points predominantly in the other direction. That's the way we force it to go to 0. So if it averaged out that it didn't point in any direction, you might expect this to be a constant, a bit like those L-function calculations in the previous lecture. But I always hate writing in this form. So this is approximately 1 plus p to the minus i t over p. So if the p to the i t's looked like they're pointing towards the positive direction primarily, then these have size typically bigger than 1. So the overall product might go to infinity. But the overall product here is going to 0. So we expect them to predominantly point in the negative direction. Predominantly is a very vague term that I'm going to make less vague in a minute. And then, so this is actually how Davenport talks about things before he goes to Maten's proof. If that's the case, if the p to the i t's tend to point in the negative real direction, then the p to the 2 i t's, afterwards the square, tend to point in the direction of minus 1 squared. But then that tells us that we should expect zeta of 1 plus 2 i t to have a pole for that to diverge towards infinity. But as I said, fairly simple considerations where zeta function tells us the only poles are at s equals 1. So this is sort of a standard heuristic to finish off the proof of the prime number theorem. And this is proved nicely using a clever identity of Maten in probably the treatments that you've read. But this is, if you look at both Delaveret, Poussin, Ademard, essentially they're trying to capture this idea in their proofs. And it was Maten came later, the proof of the cosine identity that came afterwards. So what I want to do is go back to this original heuristic and try and understand it a little better. So to do that, I'm going to make a definition. I guess I'll use the bottom of this board. So for coefficients an greater than 0, greater than or equal to 0, let me define the distance between two functions. I've done this in too much generality. But that with me would be this thing. So this is not exactly a distance function. Maybe I should just cut to the chase of the simplest example, which would be if we take a n to be 1 over p of n is a prime and 0 otherwise, so the whole thing works in some generality. And what's the idea going to be? The idea is that we have two multiplicative functions, f and g, both inside the unit circle. And this is some sort of measure of how close f and g are to each other. So if f and g were to be equal and also to lie on the unit circle, then f g conjugate would be 1 everywhere. And so 1 minus 1 would be 0, and this so-called distance function is 0. It's not quite a distance function, because if f equal g, and we're inside the unit circle, then this wouldn't actually equal 0. But there's something good about this definition is that it satisfies a triangle inequality, and that's what's really going to be relevant to us, is that, well, there's several forms of triangle inequality. Let me just give a traditional one so it satisfies this. I have a preprint on the archive quite recently, which is for the celebration of 100th or 150th anniversary of the Math Association of America, where I try and describe some of these ideas. And there's a competition there to find the best proof of this. So we don't like our proof very much. Had three entries so far, one of which is very good, I think. Anyway, have a look there if you like elementary identities on the archive. Where is the determinant of x on text? Oh, I'm sorry. Yes. Thank you. Got that one too high? There's the middle one. OK, so what's the point of this in this context? So the log of the Riemann zeta function at s is, we can think to the right of s, this is sum over the prime powers of 1 over m p to the m sigma times p to the minus i m t. So here I'm putting s is sigma plus i t. So if I look at log zeta sigma, actually, I mean, sorry. So I just want to consider this natural quantity. Then that's the same as we said before. When you take the absolute value of something, you can sort of take the real part upstairs. So that's the real part of the log of zeta sigma over zeta s. And then if we just use this formula, what do we get? Well, s and sigma have this part the same. So we get the sum over the prime powers of 1 over m p to the m sigma. And then for the zeta sigma, this is t is 0. So this is just 1. And for the other part, we have that. Now, if you look at our definition up there, it looks pretty similar. And if you like, this is comparing with the appropriate set of weights. This is m. The distance between 1 and n to the i t, all the way to infinity. So the triangle inequality will give us simply that, well, you can write that as. And then just by the definition up there, the fg prime, I can rewrite the second one as just divide out by the same function on the unit circle, i n to the i t. I can just write it like that. So in other words, this is equal to 2 times the distance between 1 and n for the i t, infinity. Thank you. So now what am I going to do with this? I'll simply square this. So I get myself in here, substitute in these. And then we get back, well, almost a slight fail. We almost get back M10's inequality. We actually get this thing. OK, so all I'm doing here is I'm squaring. So this is d squared. Over there is less than or equal to 4 times the d squared over there. I'm substituting from the line above. Sigma plus i t and sigma plus 2 i d. And when the smoke clears, this is what you get, taking exponentials. So I'll just remind you of how M10 works in a minute. But in M10, you get a plus 4 there. So this wasn't quite what we wanted. But actually, that's not very hard to repair, because I could go through the same thing again. And in order to get a plus 4 there, I could get the inequality sort of purposely taking away from zeta functions here. Sorry, twice that. So exactly the same argument you see. And then the square of the minus sign doesn't really matter. And when you work it back, that would put a minus sign in here. That would put zeta s up here. And we get a plus 4 here. Again, just trust me that the algebra works. You can verify that yourselves. But so this distance function is a somewhat flexible tool to create inequalities.  This is going to look like sigma minus 1 to some integer power to the 4, so at least sigma minus 1 to the 4. This is 1 over sigma minus 1 to the 3. So together we get a sigma minus, at least the sigma minus 1. The only way this whole thing could be greater than or equal to 1 as sigma goes to 1 is that this compensates by looking like 1 over sigma minus 1. But then we have a pole here, which is impossible. So that's M10's beautiful argument to finish off the proof. This is sort of a segue between the heuristic that everybody used and the eventual proof. OK, I can see I better not get into everything I was hoping to get into. So let me, oh, I've rubbed out what I wanted, of course. So let's go back to the beginning of this proof. It does stop in the middle, right? The outside board? Or is that just a big mistake? Looks like my work is lighter than sounds. Oh, well, what can you do? OK, so going back, we had psi of x is the integral 1 over 2 i pi, the integral from c minus i infinity, or c minus i t to c plus i t of minus eta prime s over zeta s x the s over s ds. And as we said, in order to make this work, we had c to the right of 1. So c greater than 1 plus some error term. Don't worry too much about the error terms. And the tactic of Riemann, so now we're going to get in more into pretentiousness, if you like, the tactic of Riemann is that you take this contour to the right of 1 and you shove it back to the left of the residues. It's all very elegant and beautiful. Some technical issues need to be resolved, but wonderful proof. So the new approach is, why don't we just look at this and try and bound this thing? Why do all this contour integration? Why move the contours? How big is this thing? So I want to make one sort of step that's going to make life easier, is that when we're trying to count the primes up to x, the technique we use doesn't use primes bigger than x. We want to look into that. So I'm going to slightly change zeta. I'm going to regret this later, but in the next lecture I'll have a different meaning for this notation. But zeta xs will just be the sum 1 over n to the s. p divides n implies p is less than or equal to x. So this is a nice finite thing. It has an oil of product. So in the argument we use, there was no reason not to use this thing, because it'll just give you the primes up to all the prime powers for all the primes up to x and their prime powers. So we'll work with this. That's the reason for that. Just in one second. And now let's just take absolute values here and see what we've got. So what's the absolute value of the right-hand side? So this is less than or equal to take absolute values. We get 1 over 2 pi, some sort of integral. And I'll do something with limits in a minute, but let me just say zeta prime of c. Let me change. Let me write s is c plus i little t. So this goes from minus t plus t, zeta prime over zeta. x to the c, when we take absolute values of x to the c plus i t, we get x to the c. And in the denominator, we get something like c plus the absolute value of t dt. Something along those lines, well, to a constant. So maybe a 2 here. So the x to the c is a little bit of a worry because we know that the right size of this is about x, right? So the x to the c is a bit of a worry. So let's, as is typical in these contours, let's just pick c a tiny bit bigger than 1. So if it x to the c is conveniently some well-known constant times x. So this is less than, less than. Well, there's various ways we could play this game. We've got this x, constants we're not worried about. We've got some integral, zeta prime of zeta, very complicated zeta prime of zeta. Let's just take the maximum of it, of zeta x prime. And then some integral that looks more at c is very close to 1. So this looks something like this. So how big are these two terms? Well, this one's easy. This is just like a log, maybe two log. But you'll forgive me the two. How big is this guy? Well, I'm really going to be very crude and say, well, zeta prime over zeta looks like the sum of lambda n over n to the c. Yeah, forget the it. And we'll do this for n up to x. It's not 100% correct. There should be a few more terms, but they're not going to be important. The c is more or less 1. So it's certainly less than lambda n over n up to x. So it's less than log x.  The right answer is x plus an error term. And we've lost by a log x and a log t and t. Well, we chose it to be x to the half before. We might be a bit more judicious now about the choice of t. In fact, we'll take t to be a power of log. So actually, let me be careful what I say. Let me not say that. This t might well be more complicated than that. But we've only lost by some powers of log. So that's surprisingly good for such a crude argument. So another approach to perhaps working on the prime number theorem is let's just try and be a little less crude here and see where we get to. OK. So I'm so confused with these boards. OK, I guess that one's next, if it'll stay down for me. OK, so we've done this crude argument, and we haven't done any analysis at all. So note we're integrating to the right of the one line. We're not straying into the critical strip. So the first question we should ask is if we've got a function which, let's assume z prime over zeta is a fairly benign function, which it actually is. 1 over s is certainly benign. Actually, yes, we certainly understand. What I want to understand is we know the right answer is x. So we know there must be some cancellation in this integral. So where does it come from?  Well, think about the terms. The only term that we really understand very well, besides the 1 over s, that we understand well is x to the s. And what happens to the integral of x to the s as we go through a short interval? Well, the integral of x to the s over s is x to the s over log x. But more importantly, really, for us if we go through from s0 to s0 plus what? 2 pi over log x to i pi over log x. So we go over a short integral, short part of the integral up the vertical axis, then that difference is 0. So in very short intervals of length 1 over log x, the x to the s part disappears. So there's a win there. We just have to learn how to exploit it. How do we exploit the cancellation thanks to x to the s rotating fast? So what's the obvious way to exploit that? Well, integration by parts will allow us to first integrate the x to the s, and then hopefully the rest won't get too nasty when we differentiate it. OK, so if I was working with, now I'm going to switch back to another notation, which is over there. So if I'm integrating a of s x to the s over s0, here a of s is zeta prime over zeta, then what's the idea now is I'm going to integrate by parts. So it'll be a to the s over s, the integral of x to the s, which in other words is x to the s over log x integrating by parts. And then we've got minus the two other terms. So minus the integral of a prime s over log x x to the s over s plus the integral of a of s over x to the s over s squared over log x. OK, I haven't put limits there. Let's not worry about limits. What's going to happen because of this circulating around so much is we can basically consider that to be 0. And if we go to infinity, that's certainly an identity. And actually to study this as an identity, let's just take this and go back over here. So we know this represents the sum a n n up to x. What happens in this first term here? What is a prime of s? What is the derivative of a Dirichlet series? Well, this is the sum of a n over n to the s. So minus a prime of s is the sum a n log n over n to the s. So this first term corresponds to the sum a n log n over log x. Well, I think you can guess what the second part represents. And in fact, this is sort of a standard trick in using Perron's formula. So this part actually is easily bounded. Now, assume all the a n's are less than or equal to 1, which is not quite true in our problem, but is almost true in the average of a lambda n's is bounded by 1. Then we can just take an absolute bound of this with 1 here. So the sum 1 minus log n over log x. And this thing is less than x over log x. So if I'm just putting a 1 in here, then this would be x minus log x factorial over log x. And then by Sterling's formula, the main term is x over log x. So the meat of this integral here, the thing we want, this is small. I mean, we win by 1 log x anyway. The meat of it lives in here. So let's just examine what we get out of this integral as opposed to before. So before, when I was integrating this thing and taking absolute values, I had something like the maximum in my interval of a of s times x. So here again, I'm thinking of s as 1 plus 1 over log x plus it. And I'm thinking of an integral from minus t to plus t times log t. Yeah, that's basically what I had up there with zeta. And now what do I have? Well, I have something a little different when I make the same. So this is the bound that I get by taking absolute values in this integral. Now I've integrated by parts one time, and I try to do the same thing. What is it that I get out of this thing when I take absolute values? I still get the x. I'm actually going to pull out the same thing, the maximum of a of s. Excuse me? You're going to bring your problem this time. Yes. There's an extra log x. I can bring outside. That's cool. Because remember this was bounded by log x. And I divided through by the log x. That's cool. Less than, less than. And now I've got inside this integral from minus t to t of a prime over a, in absolute value, of 1 plus 1 over log x plus it over the t over 1 plus t. So have I won or have I lost? It's a bit hard to tell. I've made it more complicated at the very least. That's good. So here, what can I do? Well, for instance, bounding this, I could calcium it. I can calcium it. Because a prime over a is kind of hard, but a prime over a squared, it's easy to play with. Terry looks like he's in pain from me saying that. But anyway, so one idea to get up a bounce would be to expand it in that way. Perhaps I don't want to get into that at this stage. But suffice to say that proceeding like this, you can, for instance, in this problem and in other interesting problems, reduce this bound to root log x from log x by this. So there is a win. Trust me on this. So this is something we're going to get into more when we get into Harper's new proof of Hollas's theorem is how you can win by doing such things. But this actually, well, I guess I'm running out of time today. This is going to come in now. So let me actually, let me finish off by something fun. So what we're going to do is we're going to revisit the proof of the prime number theorem. We're going to do it to the right of one. We're going to do it by taking absolute values, though we're going to be a little bit more clever about how we construct the integral. But we're going to do it to a slightly different function from the standard function. So OK, if you've been lost up to now, this bit is kind of fun to come back to.  And it's got a Taylor series from then on.  And then what I'm trying to get to is minus zeta prime s over zeta s, how does that start? That starts with a 1 over s minus 1 plus gamma. Plus gamma, yeah, no. Minus gamma, OK. So that's just a calculation based on this series. Now, what we want to do in the proof of prime number theorem is integrate zeta prime over zeta. And the whole issue for the main term becomes the polar s equals 1. But in the problem we're working on, we want to actually get upper bounds on the integrand to the right of 1. So somehow we want to get rid of that pole at s equals 1. So here's a nice way to do it. Let's just look at zeta prime s over zeta s plus zeta s. So that's that minus that. The pole's gone. If I subtract 2 gamma, this is at s equals 1. It even vanishes. So I want to play with this function. Now, let's see what's kind of neat about this function. Now, it has no pole at s equals 1. We've just canceled it out. But let's have a look at the zeros. So what I want to say is what are the poles of this? Now, we know that the poles of zeta s, well, the poles of this must be at least the pole of one of the terms. There's no poles of gamma. The only pole of zeta s is at s equals 1. That canceled with the poles of zeta prime over zeta. So the s equals 1 isn't a pole. And I released terms, contributed pole. What are the remaining poles? We said the only poles for zeta prime over zeta are the poles s equals 1 and the zeros of zeta s. But they remain poles here because there's nothing doing with these other terms. So the poles of this equal the zeros of zeta s. So one way to phrase the Riemann hypothesis, instead of looking for zeros of Riemann's zeta function, we can look for poles of this guy. And this is the guy we're going to put into Perron's formula tomorrow. But for today, I just want to finish off by playing with this. So if you're a good analyst, you might have good ways of tracking zeros. But if you're like me, the only thing you know that's easier to do is to show something's not a pole. Because something's not a pole means it doesn't diverge to infinity, which is relatively easy to do. So we have this function. I'm going to call this function z of s. And the Riemann hypothesis then can be rephrased, z of s has no poles with real part of s greater than a half. And what I want to do is think about a technique that would allow me to prove that. Oh, no, it's not staying. OK, well, anyway, off it goes. So how can we prove that a function has no poles in a certain ball, say? Well, one way to do that is at a certain point is if you have a Taylor expansion with a certain radius of convergence, then the thing converges. It doesn't have a pole. So how about this for an idea? Here's the half line. And I want z of s to have no poles to the right of a half. But I only want to play z of s to the right of 1. So here is z of s. And what I'm going to do is try and build a Taylor series for z of s at every point to the right of 1. And I want to prove that z of s doesn't have a pole there. Well, if I can just get radius of convergence a half for every point here, I can include every point to the right of a half. Yeah, so every point to the right of a half, I go sort of half minus epsilon, just enough to skip over the one line. And then, hopefully, it'll be within the ball of convergence there. So the Riemann hypothesis can be said to say, or Riemann hypothesis can be is implied by, z of s 0 has a Taylor series with radius convergence at least a half for all s with real part of s greater than 1. So now, suddenly, I've got the Riemann hypothesis living to the right of 1. So how do you prove that a Taylor series has a certain radius of convergence? Well, when you write it out, you write it out well, in terms of the Taylor coefficients, obviously, et cetera. So I want the s minus s 0s to go up to a half. So this grows like a half to the k. I can make this converge if this grows a little bit smaller than 2 to the k. Or actually, if it grows like 2 to the k, then I can get any radius convergence less than a half. And that'll be enough, obviously, in open interval. So here's my new Riemann hypothesis is z k s 0, z k s, over k factorial, where s is sigma plus i t.  So if I can just take these derivatives, high derivatives of zeta prime over zeta, of zeta, and well, gamma, we know how to do high derivatives of, obviously. But if the first two terms, I can do high derivatives of them and bound them this well, then I'm going to prove a Riemann hypothesis. So this is what we call the pretentious Riemann hypothesis. Again, I haven't told you why pretentious. And let me just say that if you assume the Riemann hypothesis, then just by classical techniques, Riemann hypothesis implies that log of 3 plus t. So although all we need to assume is that there's some bound with a fixed constant for each point, actually, the Riemann hypothesis will come back and give you a uniform bound at every point like that. So how we're going to start next time, I guess, is we're going to make this assumption. And we're going to prove that error term of x to the half log squared x, only playing on the right of 1. Now, that may not surprise you. I mean, we have a proof already because this implies the Riemann hypothesis. And we saw the Riemann hypothesis implies that. But I want to build a technique that doesn't go into the critical strip or use zeros. Then what we're going to do is, well, OK, we can't prove this. We can't prove a Riemann hypothesis, unfortunately. Maybe we can prove a weaker version of this. Something else to the k. And using that, well, we can do that. And using that, we can prove these other results. So that's what we'll do next time. And once we've done that, which hopefully won't take more than 20 minutes, then we'll move on to this more general technique using Perron's formula. OK, that's it. Any questions or comments? Yeah. So like, if you prove this, z equals 0. So then there is a difference from the other one. It's the plane instead of playing against type 2p, it's playing against two real factors. Can you write that as the distance from between 2p and 2p? So to prove? Zero 0. So instead of the cosine identity, you have the L1 chi 1, Ls chi 2, Lx chi 1, chi 2. Can we write that in there? The answer is, yeah, that there is a formulation. Oh, yeah. Can you do it? But it's not. Yes. It takes a little bit of explanation. So I didn't really do a very good job talking about the distance function. Maybe I will next time, and then I can try and answer that question. I mean, there are some different issues to talk about, but I'll talk about at least one of them. OK, Terry. So how do you philosophy, I guess, is to minimize the law of quantum theory? God, yeah. So I mean, I had plans to talk about how everything in Riemann's memoir comes into the proof of the prime number theorem, but the only thing really that's left is Zeta being analytic, having a pole less equals 1.


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