So, Brad, John and Justin have arrived. Okay. Good evening, everyone. I'd like to call the regular meeting of the Berlin Select Board for February 1st, 2021 to order. Additions or changes to the agenda? Brad, you'll see on the 7.15 time slot there, Phil Gentili has asked to put his his reappointment to the Berlin Conservation Commission on hold to a later date. Okay. Seeing no other changes, public comment? I have one by Peter. I'm Peter Schover. I live on Brookfield Road. And along with Susan Rich, we own the 25-acre Chapman Churchill Woodlot on Irish Hill, adjoining the townland on two sides and adjoining the Rich Farm on the third side and the Schober Irish Hill Farm on the fourth side. So that's where our interest is and our immediate concern should the town grant a permit for vast that it be clearly written in writing and to include signage not to direct vast snowmobiles onto adjoining land. Unfortunately, there are maps in circulation today of Irish Hill, including some USGS topo maps that indicate trails by a dotted red line over most of the townland and onto the Chapman Churchill Lot, which has no public trails, and on down onto the Rich property down to Brookfield Road. So those maps would have to be used very carefully to instruct vast or anyone else where they can go and where they shouldn't go. I just mentioned that as an FYI because those maps can be misleading. I think there are broader concerns, safety concerns, should a snowmobile permit be issued, both at Brookfield Road where what I call the Lee and Stewart Road, it's the east end of Darling Hill Trail, a joins Brookfield Road and where the small parking area is. That area can get congested very quickly. There's a blind hill there, and if any additional parking were permitted, especially in the wintertime when the road's narrow, it could cause a really serious safety issue at that point. The other concern that I would have is a more general concern of having pedestrians, snowmobiles, I'm sorry, snowshoers, and cross country skiers on the same trail with snow machines on some of the very steep narrow sections of the trail, the tower trail, if that's the intent, the tower trail is of course the steepest and the narrowest. I wanted to note that the the Berlin advisory survey, I think fairly clearly indicated that the voters were not in favor of snowmobiles on town land, and I wonder how much weight the select board puts in that survey, and it would be my suggestion that perhaps this issue be tabled until more investigation can be done. The issues itemized and hopefully resolved with vast to everyone's satisfaction in preparation for next year. I don't think that trail meets any of the Vermont town forest trail specs or probably vast specs, but that's an issue that hasn't been determined and I think should be on that list. Thank you for your time. Thank you Peter. Any other comments? I just want to address Peter's, one of Peter's concerns, and Peter we've been having these discussions for a couple of months now and I think everything that you've mentioned has certainly come up and needs to be weighed and needs to be worked on throughout the course of the next however many months before the next vast season if the board chooses to move forward. So everyone's following the reason I put the action item or requested the action item beyond there tonight or possible action was because after the public hearing I figured out exactly who had the ball and what the next steps were. So my intent of the possible action was to talk about the public hearing with the select board and depending on how the board felt to offer a motion to have the conservation committee work with vast to mitigate some of the very concerns that you brought up whether it be the steep part which I think is an issue the multiple you know types of use of the trail and it'll give the select board a little bit of time to work on the the road issue and try to mitigate that issue as well. It was not to push a slum of the chain trail through this this season it was more of trying to figure out who had the ball and make sure we're all on the same page. So with that I'll be quiet. And I'd also like to apologize. Can I respond to that? Sure. Thank you Brett. I think that would be an appropriate scenario for the Berlin Conservation Commission to address with vast as the applicant. My question would become what happens if there are issues that appear to be unresolvable? Do you have an example? Well yes I think the steepness would be one of those. I think that's mother nature that's not about the change. They couldn't change that with bulldozers and blasting when they put the new tower in there. So you know I have to say that I'm personally aware that snowmobiles can travel that trail. Not recently but but vast tends to want to groom their trails. I'm not sure that they would be able to groom those trails and they tended to travel in groups and that of course presents more issue with town folks that are using the trail for other purposes. So I just think you know it's possible that they could come to an impasse and then do they return to the select board and say what do we do now? It sounds Peter I'd like to personally apologize for not responding to your email. I feel like you sent one. I do apologize for that. I think that what John's the point of that the action item on the agenda is to actually get into that type of a discussion and figure out what we don't what we don't have in place at this particular moment is an actual plan of action should something other than snowmobiles arise or another outdoor recreation. And certainly depending on where you go you know sometimes you can coexist. Sometimes you can't probably on some of these trails. I completely understand that. So I think what the town needs to do moving forward is have a strategy and a plan in place on so that when when this is all the dust settles however this may turn out that the next thing that moves forward can move forward in a productive smooth manner where everybody knows the process and procedure. I think that's that's the biggest thing behind this as well. Thank you Justin and a plan a plan is is a good thing. Any other comments? I have a comment. Yes. I understand all of the comments that were that were made but I'd like to bring up something that is less quantifiable. But as far as I'm concerned and I think a lot of other people it is it is a real concern. And that is the the idea of or the product of noise that the snowmobiles make. The proposed trail would be above us. I live on the end of Brookfield Road. We my wife and I have property on both sides of the road. And it is certainly possible that we would be affected by noise that the snowmobiles make and they can especially in large groups that are constant. It can be an incredibly annoying thing. I'm even more concerned that although I understand the vast organization and its rules and regulations but very often people take they veer off of those trails just to go elsewhere. It'd be very difficult to police and the most likely place they would end up is right on our property. You're very close to it. So that noise I don't know how you guys address that but it is a real concern of myself and my wife. Thank you. Thank you Mr. Hurley. Any other public comment? Yes Ms. Drysdale. You're on mute. There. There you are. Yep. Okay. Well I wrote sort of a letter this afternoon that I'd like to read. I've lived on the south end of Breland Pond for 41 years and I enjoy walking the Darling Road Trail and Mountain Bike Trails with my dog several times monthly. I'm also an avid cross country skier and have shared snowmobile trails with riders since before vast existed all over Central Hermione. Occasionally making a donation to a local vast group as a courteous gesture. As has been noted, riders almost always slow down appropriately when we meet in a friendly way is tempting for me to support the vast proposal as I'd love to spot a car and ski from Northfield to Darling Road and then across the road across the Pond to Home. But after mowing about this for the past few weeks, I've decided to register my opposition to the proposal and to urge this elect board at least to look further into the issues brought up by the conservation community and other Breland citizens. Three of these in particular are a particular concern to me as a frequent user of the area. One, Butland's conservation. There are three places on the Darling Road and the Tower Trail that are very wet much of the year with swampy areas on both sides of the road. Two, steepness and safety. The Tower Trail is really very steep and rocked for long periods. I'm familiar with the other vast trail to Northfield from the Breland Pond Road near Exit 5 which has steep spots but this is worse. I'm also very concerned about snow machines using black and revealed roads on a section that's narrow, steep and dangerously blind and which frequently is used by walkers, runners and bikers even in the winter. And three, popularity. Since the pandemic, these trails have been discovered by many more people, singly and in groups, searching to get outside and exercise safely in peace and quiet. Many are accompanied by dogs who are rarely leashed and joyfully romp ahead of their family. I expect even when the pandemic is over, this area will continue to be a winter haven for Central Vermont walkers, skiers and skiers. I fear that if this trail is open to vast, it will be very popular where we snow fielders as the short cut to the Northfield Bear A.S. It is a journey in a non-mortarized focus to be seriously compromised by frequent encounters with snowmobile-ers even at once. Thank you. Okay, thank you. Any other comments? Brad? Brad? Yeah. I have a Galaxy Tab 2019 have raised their hand. Yes, this is Shea Miller. I live on Black Road and I just would like to voice my opposition to the plan to allow snowmobiles. I have concerns about having snowmobiles on the roads and trails. A lot of the users there are walking with small children and pets and these higher impact uses are definitely going to negatively impact these users. I also have concerns about the environmental impact and the fact that this plan seems to be moving forward without really a lot of assessment about the negatives it's going to have. And I do have concerns about the noise even under the current situation just where the vast trails are on my end of the pond. I already can hear the snowmobiles so I do have concerns about them being able to have access on the roads as well. And that's the end of my comment. Okay, thank you. Brad, there's Theron Lay Sleeper has a hand raised. Hi everybody. My name is Theron. So my understanding is that the town forest is or was initially designated as a place that was meant for non-motorized recreation, sort of a natural habitat conservation and also a resource for the town as such. And I may be mistaken but that's what I understood from the last discussion when we talked about the proposal from Vast last time. And I think that that's a very valuable thing to have in town that is available for everybody to use as a space where there's quiet, there's fresh air, there's habitats for animals and it's a place also where people can walk their dogs and recreate. And my understanding is also that there are existing vast trails that go between Bury and Berlin and Northfield, a little bit further to the east it looks like, the southeast. And so factoring that in it seems like this would be a nice shortcut for the snowmobiler as it would make a loop. But there are lots of other places where snowmobilers can travel and get to where they are interested in getting to. But I think it would be it would be a greater negative impact for the other folks that wanted to use this land for recreation and non-motorized recreation. It would be a benefit for the vast trail but it would also negatively affect all these people who are interested in keeping it as a designated quiet and natural area. Like Theron, I just want to try and end there and clarify a couple items for you. With this land from the research I've done just so you're aware, the town acquired the land in, it was the 28th of December 2000 and we actually received the deed from the Vermont River Conservancy. I researched it for restricted uses in areas that can be utilized and under the restricted use from day one, snowmobiles have always been. If we read here, it just says number seven, there shall be no operation of motorized vehicles on the protected properties except for uses specifically reserved, section three below, such as wildlife forest management, trail grooming, maintenance, handicap access, and for safety and emergency purposes, snowmobiling may also be permitted. So, it says specifically in the deed. So I just wanted to give you that clarification. That's all. I just wanted to throw that out there, so everyone. But we didn't hear it. Sorry about that. Can you hear me now? Okay, perfect. Back when we bought the property in 2000, with the deed that I researched, basically the shortest snowmobiles were permitted through the Vermont River Conservancy when the property was bought. So I know that was some of the question that was out there, as if we were going against the original purchase, the deed, things like that, in consideration of this. So I wanted to do my due diligence and research it and specifically emergency purposes. Justin, as soon as you stop looking at the camera, we can't hear you anymore. Sorry. It's okay. Yeah, I guess I better get another camera if we're going to continue to do it this way. The deed allows for snowmobile travel. That's what it boils down to. When we purchased the property in 2000 from the Vermont River Conservancy, and I just wanted to make sure that it did or it didn't, and go from there. I just wanted to share that. Okay, thank you. Brad, Ron Lyon has his hand up. Okay, Ron. We lost him. Wendland? I just want to clarify about the easement, and I can provide it for anybody who wants to see it. Our current easement is with Vermont Land Trust, and it does allow for snowmobiling provided that we can show that it meets with the primary reasons for having the easement, which are conservation of the property, and the second one is for it to be compatible with non-motorized use. So we have to ensure that those two things happen, and if we can ensure that, then we can allow snowmobiles. And if anybody wants to see the current easement that we have, I'd be glad to share that with people. Okay, Ron. Thank you. I think I'm back. Here we go. Am I back on? Yes, Ron. Sorry.  Thanks for inviting us all tonight. I had some comments, and one of the things that I guess I feel is that there's a path forward on this, and I think that's what John is saying. The path forward to me is we've got a very important resource for the town and for the people in the town. It I think is really just very important to take the time to understand it and it's necessary to really understand the proposal and the impacts. And before we approve it, I think any steady any motion or any movement would have should take into account that it would be subject to and after review of many different aspects of the process. Some of the things that I think that will result in is a much better outcome for the town and all users, residents. Some my concern certainly would be that all the environmental issues, this was one of the key, that's the key in the whole town plan, the key conservation area. And it does have a lot of habitat and steep slopes and erosion issues. Some of the things that I think I'd like to recommend the town look at closely is, you know, what is the use of finding that use? What are the hours? What's the speed? What's the location of what we're putting in the shared use? How do you can we make it acceptable to those people now that many, many residents enjoy it? And through this steady come up with this program that says, yes, we can do it or no, we can't do it. And then address some of those issues. I think noises are concerned just because of the valley. And we've experienced the same things with snow and veils, testicle, wetlands and erosion control and vegetation and habitat, all those things that are conservation related. I think if we look at the parking and the roadways, we've had some residents that have expressed interest in all those things. And I think it's important that we look at them before decisions made and put that process into place that we've got a good conservation commission. We've got really sharp people in the town, good planning commission, good slight board. You look at those aspects of the proposal prior to making the decision and do a real real honest public input public participation process to say what do we want to do as a town? So some of those got a long way of saying we really need to look at all those aspects. There's no rush. It's like, you know, I think our planning commission was like one once was really good reviews of the town center over a period of the years. This won't take that, but it should take the same considerations for doing a good job of making decisions. So I'd say we shouldn't we should do this study. We should have this conservation people involved. The land trust involved. Certainly the snowmobile is involved fast as a great organization. So there's lots of ways within the next year, you could come up with some really good decisions and good firm decisions.  So that's kind of what I was looking at. And I think John is heading in that direction. The path forward is good. And all these comments, I'm just pleased with the comments. I walk that trail fairly often. We're in summer with and without dogs. It's just what a resource for the town at this point. So so just doing a really good job. So we get it right. I think it's important. Thank you, Ron. Brad, Trevor Whipple has his hand raised. See you Trevor. Good evening, folks. And I don't want to be redundant to what I said during the last meeting, but just want to make sure that it's on, I guess in the record, I would be opposed would ask that before any decisions are made that there be, you know, a thorough assessment of we're talking shared use, as I mentioned before, not just of the path. So shared between foot and walkers and hikers, but also the shared road. And would also ask, and I have not asked this question internally. I don't want to have any conflicts, but ask, does this have any bearing on the liability to residents as well? If we're starting to have this real blend of uses on pads in the woods that now will potentially change use as well as a town highway. Also looking at what could the potential impact be to the town budget? Is this going to require any additional enforcement, any additional response by the police department? Any additional response by our rescue folks? We do know that we read regularly about snow machine accidents just like we do car accidents. So would just ask very much in favor of a lot of what I've heard this evening and I was on the phone earlier before I came in by computer, but would ask for a thorough assessment by the board as to what the risks, the potential cost is. And what's the gain? This is an assessment of what's the cost versus what's the gain to the taxpayers in particular. This is townland and what's going to be a best benefit for everybody in Berlin. So I just, I thank you for putting so much into this and having opportunity for the town to weigh in and just ask that we really think about all the different angles here. Thank you. Grant, I have an L. L. Miller raise their hand. Okay, Mr. Miller or L. L. Miller. Hi, this is Lindsay Miller, Berlin resident. I had hopped on the last call that the last discussion we had about this, but I wanted to join here and also, you know, make note of my opposition to this plan. I think aside from just the long-term impact that putting in a new road at a high elevation through sensitive habitat has, I think ultimately there is a way that we could have vast trails and snowmobile use in the area. You know, I think there's just using a little bit more thought and maybe being a little more mindful about it. Maybe the path can be routed elsewhere, maybe through an existing road or an existing, you know, section of different vast trail.  So again, just wanted to throw my name as opposing this idea as it currently stands. Thank you.  This is the fifth full month that we've discussed this. Well, we're going into the sixth month. We haven't made any decisions. We're still collecting public input. So, you know, when the survey was put out, only 59 people total responded and just barely over half said no. So we're talking about 20, 29 or 30 people saying no total. And a lot of them are on the call tonight. And we certainly, you know, want to address a lot of those issues. A lot of them we have in previous meetings and Vastus Command and talked about what we're going to do around, you know, where people would park in the signage and things like that. And we're happy to keep listening and to make sure people get their answers and feel comfortable.  This is Rob Griffin-Suter at also Berlin resident just reiterating my opposition as expressed in the last public hearing. In particular, because of being such frequent users of the trail in all seasons of the year, including winter, and a strong sense that in this case, the benefit to Vastus of being able to use that trail does not outweigh the disadvantage to all of us who use it for non-motorized purposes. So it would urge that that very strong consideration on our part be taken seriously by the board. Thank you. Any other comments for public comment? Okay. Hearing none, we'll move on to Treasurer's Report. Diane? Okay. The first thing I have is the contractor's application for payment. I put that on the desk and I've sent it to all of the SelectWord members. This would be the seventh payment and I would like to get it approved and then have the chair sign it. It's for $252,950.58 and it's for Dubois construction for the Payne-Turnpikemore wastewater improvement. Hear a motion? I'll make a motion. Make a motion to approve the, you said seventh payment, Diane? Yes. Payment number seven, yeah. Payment number seven to Dubois construction. For $252,950.58. I'll second the motion. Thank you. Any further discussion? All those in favor? Hi. Hi. Brad, Brad, do we have a quorum here? Is it just, well, I guess we would if you're on two. Who else is on from the SelectWord? Well, Flo's in here somewhere. Flo? Yes, hello. I would have made the motion, but my husband was on the phone, so you would have heard him talk in the whole time, so that's why I allowed someone else to do that. Thank you. Are you in favor or against? Motion carries. What else, Diane? Okay. What I do have is for 2020, I got a report from the third party credit card company that people use to pay taxes and utility bills, and I want to let you know in 2020, we had 300 transactions for a total of $179,297.47. So every year, more and more people are using the credit card. I do believe a lot of it has to do with COVID at this point. That's why it's probably as high as it is, but it's just to let everybody know that it is working, is working well. Okay. Thanks, Diane. Thank you very much, Diane. Yep. Let's see here. The Newtown Center application. I just, I sent out notice to the select board today and various other committees that the Newtown Center application has been filed with the state. The expected hearing date is February 21st, and I believe it's at one o'clock, but I need to confirm that. So the, and I looked at our, on our calendars, it's really started this formal process in 2014. So it's been a, been a while, and it's very, I think it's a very good application. I encourage everybody to go on and look at that application. It's also, Diane has put it on the town website. And so I'll be garnering support from the town on that hearing date. If there's any questions, I'll take the questions. Hearing none. Thank you, Tom. Thanks, Tom. Okay. Development of Route 302, Town Fair Tire, the old Steakhouse location. So I'm going to share my screen, Brad, here. So at least I think I'm going to. We can see it. Very good. So in our, in the Berlin zoning regulations, there's a provision I've had, and I have it highlighted here. Well, I have it highlighted here that any applications that front on Route 302 in the commercial district, which this application by Town Fair Tire does, that they provide sidewalks. And you could see on, on Section E, the town of Berlin has the ability to require the property owner to be responsible for the maintenance of sidewalks, irrespective of whether they are located within public right of way or not. And so I've talked to the Town Fair folks and they have submitted for the Town's review and approval and agreement to where they, in effect, will take care of the maintenance of that section of sidewalk on their property. The development review board is meeting here on February 16th. And I know this issue will come up with the DRB. If I'm, and I'm making, making the assumption that the Town of Berlin does not want to do snow maintenance, snow plowing, and other maintenance on sidewalks at this time. And that if, to move this project forward then for Town Fair, I'm recommending that the select board enter into the screen with Town Fair to allow them to take care of the maintenance. I sent this to the select board in their packets earlier. Tom, I didn't see any, I guess I didn't get this one. There was a bunch of different information we received. But do we have, do we have requirements on what the maintenance of that would look like, set in anything anywhere? I mean, what is the maintenance plan, just in general? We do not say that in our zoning regulations. It could be a condition of any permit that is granted. So just let me know if the select board has a desire to have any such condition of that permit include what the actual, if it's a, you know, snowfall amounts or things like that. Yeah, I just think the fact that they're going to maintain it is pretty vague without a standard to stick to. Well, the DRB would be open for that. The DRB would be the one that would probably want to make that recommendation, wouldn't you think? I think they'd really rely on the select board, to be honest with you. Something that I could find, see what other Towns are doing that have this. And I could send that to the select board for your review and see if that's what you want to permit, any permit condition to state. This time our, the agreement for the Town Fair to put in and maintain a sidewalk, when does this need to be signed? Well, the DRB is meeting on February 16th, the first hearing. This, likely this project would take two hearings. It could be done as quickly as one, but I would think that it would take two hearings. So their, their second, their next hearing after the 16th of February will be March 2nd. So I would think by March 2nd, this would be concluded. So if the hearings, the 16th, we have a meeting on the 14th. Correct. So we have a couple of weeks to get our ducks in a row and you have the meeting on the 15th, right? The 15th. We have a chance to get our ducks in a row and have a set of standards that we could follow. I believe they would do that, yes. Okay. So I would say put this on the agenda for the 15th and we'll take in, the select board can get their comments into you and you can make a list of, what we feel would be a acceptable maintenance package. And so look at it, look, look at through the window of if, if the town was doing the maintenance on it. So yeah, but I'll get that to you. Yeah. Anything else on this? No. Okay. Just real quick, just with the maintenance, you said look at it from the town, just Tom, when you send whatever it is you're going to send over to us. Can you make sure it includes, I mean, like future deterioration or repairs and that to some level? Yes. Thank you. Okay. Anything else on this? If not, Green Mountain Power worked in the town right away. Project number 176237. I sent you in advance the working in a right way with the Green Mountain Power has sent out. They have a powerful that they need to shore up and they, and they've asking permission of the select board to, to work on that right of way. It's near 1302 Chase Road. Motion to approve the right of way work by Green Mountain Power. Second. Any other discussion? Hearing none, those in favor? Aye. Okay. Motion carries. Thank you. And thank you, Tom. Let's see here. The town clerk mailing the ballots. Rosemary needs our vote on this. Yeah. In short, the she sent out postcards to Berlin voters and subsequent to that mailing, the school supervised union, two schools, two towns in that supervisor union voted to mail all residents ballots. And so the town clerks are being told by the secretary of state that every town needs, excuse me, needs the mail ballots. Rosemary doesn't have the right to do it to mail ballots to all residents without select board approval. And so that's she, she put this document together and is requesting that the select board approve it. I'd make the motion to approve. I mean, I've talked to Rosemary about this. She, they, we got 400 absentee requests and the school board is already sending out ballots to everybody. And she, she feels like we might be able to get some reimbursement maybe from the school district if we mail them at the same time as well. And ultimately, it doesn't make sense to mail them separately. But I would make the motion to approve that we allow the town clerk to mail ballots. Any other discussion? All those in favor? Motion carries. Oh, Fisher Road, Fisher Road, Colbert financing time. So I, I still have not got an answer from the state infrastructure bank on if we can submit an application for the Fisher Road financing because the financing requires a positive vote at town meeting day from the, from the town residents. So, so we are moving along in the process. The Robert Clark with Otter Creek Engineering believes that this week they'll finalize the design and I'll get that to the select board here prior to anticipation is prior to the next meeting. So you'll see the final design on that. The, as soon as the design is done, we will submit permits to the agency natural resources to allow us to work in the wetland area as soon as that that can occur is May 15th, 2021. Robert doesn't anticipate any issues in, in getting those permits. So it's just, I think the, the key thing here is, is securing the financing and, and that's what we're going to need the town vote on. Okay. Any questions for Tom on this? Hearing none, we'll move on to certificate of highway miles. I sent this out. They recall last meeting we discussed about the black road section of maybe adding to this class three roads. Just, just, and I had several conversations and just believe that it can't be added this year. We need to, we need to follow up with some additional items to V trans. And I think that can be done. Brandy Mary call, we had a just conversation second had a conversation several meetings ago about the trans wanting to, to the seasonal portion of Rawhill to make that portion of class four. So we were looking at how to do that. It's basically a public process. And I think it's best to, when we do that process, we do the black road process simultaneously. And then cure all those issues and get those submitted. Okay. So, so I would, I would recommend that, that the select board approve this certificate of highway mileage here tonight. Tom, does that certificate of highway mileage include or not include the section of Rawhill Hill at this point? It does include it. Okay. And when is this, when, when's the like drop dead due date on having to send this certificate? And I didn't get a chance to speak with the state yet. It was my intention to try to talk with her today.  And you could also have sections of class three roads, according to the state statutes that you didn't necessarily do 100% of winter, winter maintenance on. Now, maybe that's something different. But I was going to talk with that. I think it was carry maybe at the state. Correct. So it's February 20. Justin, so, so is it possible, is it possible that we can just approve this, get a little bit more information and approve this at the next meeting? And just make sure we have just so I mean, it's probably too late. I can't hear you, Justin. I just didn't know the, if it was possible to wait, or if it was going to be a tremendous burden to put it off for the next meeting until maybe we have all of that information. I don't think you're going to get anything by next meeting to change what, what this form is. It's, it's anyways, but you could So you don't think the public hearings that we had would have satisfied it or anything in your hind, in hindsight? The, but there's a provision and then I shared it with you that within 60 days of having those hearings, you needed to be a recording with the town clerk, and that did not occur. Right. So I don't know, I don't know how you can get around that. So I'm just suggesting we just, we just do the hearings again. It's, it's one tenth of a mile. Right. And we can inventory all of our other roads at the same time and make sure there aren't any other ones that are out there. And it's minimal, but I just had a principle was thinking about it, but I'd make the motion to approve the highway mileage certificate for February 10th, 2021. Can I second that? Any further discussion? Hearing none, those in favor? Aye. Motion carries. The Town Administrator Hiring Committee. So I sent out earlier this evening my recommendation on on the town administrator hiring. I have received Concentra occupational health has done a pre-employment physical on Mr. Conti, and he has, he has passed that physical with no limitations to activities with respect to the job duties. The, I've talked to several of Mr. Conti's references that have given him glowing reports and on his management styles. The, I received late this afternoon the background check from our third party folks that do our background checks, DT enterprises, I believe is the name of the company. And, and there are no red flags on that background check that would preclude Mr. Conti's hiring. The, the only item that is missing is the results of a pre-employment drug screening Mr. Conti has, has appeared for the drug screening and as the sample has taken, we just have not received the results. So I, I, my recommendation that the select board would, would be that we, that you folks approve Mr. Conti for the town administrator's position contingent on the, the satisfactory results of that drug screening. And that would be effective hiring date of February 15th, 2021. I'll, I'll make the motion to hire Vince Conti for the position of Berlin town administrator with an effective date of February 15th, 2021, contingent on satisfactory pre-employment drug screening results. Second to motion. Any discussion? Hearing none, all in favor. I, John and Flo. I said aye. Hello? Aye. You didn't hear me. I said aye, but you folks didn't hear me. Thank you. Okay. Uh, motion carries. Um, moving on to liquor. Congratulations. Is he still with us? I don't think he, I don't know. Yeah, he is. Congratulations, Vince. There he is. Congratulations. Okay. Uh, liquor licenses. We have, there's eight licenses that I sent you folks in advance and it's, they're detailed on your, uh, yeah, on your agenda. These are, these are all the standard liquor license. I mean, these guys get there. They're nothing new. There are all the people that are constantly getting their liquor license, correct? Yeah. That's my understanding, Justin. Yes. Do you have to go, you have to go into their liquor board or something? I, I'm not, I'm not familiar with. I don't, I don't understand why we, there's no appointments to a liquor board. I'm still not sure why we exit to a regular meeting to go to a liquor board that doesn't exist. The, I think it's, uh, I think the liquor board has to, or a liquor board has to approve the, uh, licenses. I don't, the select, doesn't the select board act as the liquor board if there's not an appointed liquor board? So that's why we, the select board wouldn't have to go and leave and go into the liquor board. Yeah. Well, I don't think it really matters one way or the other. Just the, uh, just, uh, get the, uh, if you want to approve them, get them approved. Just looking for the right document now and I'll make the motion. I'd make the motion to approve the first and third class liquor license for the wayside restaurant, second class liquor license for Maplewood Limited, first class for GC Pizza Hut, second class for Jolly Associates, second class for Price Chopper, operating company of Vermont, second class for Thomas Fireman Garden and second class for Kenny Drugs, number one. I second that. Thank you, Justin. You haven't heard any complaints from the chief about these? No. Any further discussion? All those in favor? Aye. Aye. Uh, motion carries. Okay. And discussion of vast application to use town lands with possible select board action. John. So, yeah, so I'll start. I said it in the beginning, but I'll say it again. After the public hearing, I didn't have a good recollection of what the next steps were. And I thought that we should talk about it as a board and figure out. We got a lot of feedback during the meeting. There were a lot of people from both sides there. We went over Josh Walker's 100 signatures for the vast trail of people that were in favor of it. And then we went over the survey results of that the conservation committee put out on front porch forum and got 59 people back for. So we went up, we went through all of that and ended the meeting, but we never talked about what's next, right? So I don't know where people stand. I was going to suggest making a motion to have the conservation committee work with vast mitigate concerns and have a draft of an updated management plan by first for another public hearing for everyone to consider and to talk about again at that point. But that would give us February, March, April, May, four good months to talk about not only signage, noise, the road and how and if we can mitigate the road issues, the steep trail issue, which I recognize it is a little bit more of an issue than it is in some other areas because other areas, such as in Norfield, you don't have as many walkers up there on some of those trails. But I think they can be mitigated. And I think if the conservation committee, the select board and vast work together, that we can come up with a plan and do another public hearing at the beginning of June. That's where I was thinking we had had, but I'm open to any other select board members thoughts and if anyone has any different ideas, all for it. John, I chime in and say that I heard a lot of concerns around the shared use, the noise, and just cutting in new trails on it. And I think that based on everything I've seen, the shared use, I've just started actually using a snowmobile and I was up on North Street and I must have come into about 70 cross-country skiers and sharing the trail. And they also had a section of the trail that the cross-country ski club was using their own snowmobile to groom the portion as well so they could ride on the trail with the snowmobilers. I think sometimes there's some fear in the unknown and so I think a well-organized approach where we have clear direction and a plan in place on how we can proceed and mitigate a lot of these issues makes the most sense. I do know the decibel levels and noise is another concern, but the reality is just like cars, snowmobiles have decibel limits and they're actually 10 decibels below what an automobile is for inspection. And I just think everything that we can mitigate through all of it and if we can't then we'll have to come to, we'll make a decision. But I really think that in the interest of the people that want to coexist and utilize that trail, that also should potentially probably have the right to use it. We should develop a plan and move forward like you were saying. Well, and it's such a great natural resource for us with the town that, you know, I think, you know, personally I'd love to see people using it. Not everyone is going to use it the same way. Some people hunt on it. Some people use their ATVs on it. Some people want a steam machine on it. Some people walk on it. Some people ski on it. And I love it, you know, and I think everyone can coexist. I think there just needs to be rules and I think there needs to be expectation set and we should, you know, try to mitigate risk and public safety hazards and conservation hazards where we can. But I think if we work on it for the next four months and we set a date, because if we don't set a date, we'll be another six months into this still kind of having the same discussion on, well, what's next, that we can actually, you know, figure out what we're going to do here. And I know that the vast trail or the vast organization is willing to put in the bridge at the beginning of the trail this spring. So, you know, I'd like it all to follow in line. I don't want them to build a bridge and then, you know, have us say, yeah, well, we decided not to at the last second. I'd like us to work somewhat on the same timeline. So if vast doesn't put in a bridge, if we never intend on having them use the trails. I think that's fair. You know, and one of the things I don't know who made the comment, I wanted to mention this too and it kind of jumped out at me as people walk the mountain bike trails. And I mean, I granted it's not necessarily a motorized sport. So it might be viewed differently depending on what your background is or what your hobbies are. But I mean, that's an example of just things that are already moving up there that are fine together and they're known. So that's a big piece of it. And I completely agree with the bridge. And in fact, I feel that bridge, you know, with the safety concerns, with the width and the actual structure itself, I appreciate the fact that we may be able to find a kind of a public-private partnership to help in times like this and help keep the cost lower. Brad, what are your thoughts on the trail or on the bridge part of it? Whatever you want to talk about, Brad. I do agree with the assessment that it is public land and the fact that there is no taxes paid on it. And so everyone in town should have the right to use it. I understand the concerns of the people who are using it now and want to continue using it as it is. I haven't walked those trails. I don't know just how steep the part of the trail is toward the top. I'm assuming that the hikers are able to navigate it without any trouble. But then again, if it gets icy, sleds, if they're properly studded, they won't bother. But then again, you can have some other troubles. I right now would like to take and hear from the conservation committee or someone that can take and tell me just how many accidents there are on snowmobiles between skiers and snowmobiles on the vast trails. I'm assuming that these trails are, if they're police, then there are probably records of that.  Brad, Brad, I don't know if you've had a chance to look yet, but there was an email that outlined some of those statistics that went to the select board. I believe it came in the middle of this, middle of last week. Maybe it should be in your email. Yeah, it was sent to the conservation commission at the select board. Okay, so Brad, what I'm hearing from you is to me, and correct me if I'm wrong, but it sounds like it's almost the same thing. Where you're not opposed to it, necessarily, you're in favor of due diligence, and that provided we can mitigate and go through any of these issues. I can't hear you, Justin. What I'm hearing you say, Brad, is that you don't disagree with the idea, and that we should hear from the conservation and move forward in research, and making sure we can work through any potential issues that pop up, which I believe is essentially what John had brought up for a point to begin with. Am I correct? Pretty much. Any other comments? You had two things there. The second thing was the bridge. Did you have different thoughts on the bridge? Well, on the bridge, if we can get the, if we go forward with the vast trail, I do feel that the bridge should be improved simply because it needs to be improved for the snowmobiles. It doesn't need to be improved too much for the hikers. So I think that the vast should probably put the bill on that one. However, if vast, or if the vast trail is not put in, I do believe that the town needs to improve that bridge so that rescue vehicles, or at least a side-by-side, can get up there to rep for anybody who has an injury like there was this summer. So I think the bridge would need to be improved to about the same quality their way. Either way. I would say. Less and, yeah, yeah. But as far as the hiking part of it goes, I don't know who it was that injured his knee up there, but you know, it was, I think I heard like three hours to get him down out of there, and I don't think that's quite acceptable. That was a huge red flag for me personally, safety-wise. Thank goodness it was only a dislocated knee and not something- Obviously it wasn't your knee. You know what I mean. Come on now. Anything else on this? Well, we haven't heard from Flo, so I'm just kind of- Okay, Flo. Thank you. I was just waiting. I started to speak a couple of times, but I wanted to make sure you all had a chance to get your views in. I agree with what you've all said. I'm on the fence right now. I'm not necessarily in favor at this time. I think there's been a tremendous amount of opposition, and I want to look at that more heavily. At the same time, I'm not opposed to more research, but at this point in time, I think there's more opposition than there are people in favor. That's my take. And at the same time, I'm not opposed to going forward, doing additional research, but right now I'm not totally in favor, based on what I've heard both sides. Anything else on this? Hearing no more comments, we'll move on to minutes. Mr. Chair, I did not receive minutes in- Well, hold on. All right, Justin. I thought we were- Can't hear you, Justin. Do we need to do an action item? Do we need to have an action item to do the research or to have set a timeline for when we'd like to have the Conservation Commission have the plan in place? Well, the Conservation Commission, they have to- One of the things they need to do is see if they can change their plan. I mean, with the land- More than that. They've already mentioned that they can. So I would feel like we would want to have some model moving forward. If we just move along with this, we're not going to get the answers to the bridge and the funding for that or any of these other items moving forward. So I feel as though the Select Board should request a timeline, since we have heard from the Conservation Board that they can submit that to the Land Trust and that it shouldn't be an issue. So I'd like to see- I mean, I'd like to see what that looks like because what I don't want to do is if, for some reason we get in favor of, we decide to move forward as a board or the new board or whomever that will be, I don't want to run into a situation where we don't have anything rolling forward and we just can't do it again because we were on our heels on this one. So. So I'll make a motion to direct the Conservation Committee to work with the VAST and the Select Board to mitigate concerns and have a draft of the updated management plan by June 1st, 2021. Wendy? Is that okay for the discussion, Wendy? Yeah, we will need to talk to Vermont Land Trust because we independently cannot put together the management plan. There's certain criteria that we need to be following, but yes, we can move ahead. This is what I wanted to do is sort of be able to have a list of concerns and a list of things we need to research and see where it goes from there. We have some environmental things that are mandated we need to look at. You know, there's multiple things that need to be looked at. I don't feel that this is a guarantee that it's going to go through. We really need to do the research and make a decision after we've done the research. John? Yeah, so we're in the middle of a motion, Brad. Yeah. The public hearing was in the, or the public. Okay, I thought I was asked to speak. I'm sorry. It's fine, Wendy, I just wanted to make sure we don't go back to a full public hearing in the middle of a motion here. No, but historically, we have taken under the comment section or we've always allowed people from the audience to speak. Yeah. Okay, so any other comments on this? Yeah, John, did that motion include a timeframe? Yeah, it was June 1st, 2021. Just wanted to reiterate that. Thank you. No further comments. All those in favor? Aye. Aye. Aye. Motion carries. Is the motion 4-0, Brad? What's that? Is the motion 4-0? Yeah. Yeah. Or is there, okay. Well, I mean, realistically, there's absolutely nothing wrong with having, finding more information and having a discussion. Right, right. And that's all I'm asking for is just, so we're all on the same page. So it's documented that, you know, by June 1st, we'll come back and have another public hearing with the Phoenix. Now, thank you. Yeah, to me, that makes the most sense. I was trying to think of when we should have a public hearing, but June 1st is as good as any day. The rest of it, I mean, just the collecting of the information is, I think is going to take a while, but that's just me. Okay, moving on to approval of licenses, permits, vouchers, and applications. I have that in front of me. I'll go ahead and do that now. I make the motion that we approve payroll warrant 21-15 for payroll from January 3rd, 2021 to January 16th, 2021, paid on January 20th, 21 in the amount of $38,460.94, also payable warrant 21G16 with checks 2869 through 2898, in the amount of $94,483.54, also the December general journal entries. Thank you. Here a second? Second that. Any further comments? All those in favor? Aye. Motion carries. Let's see here. Tom, you said there were no minutes, so round table falls. No, not tonight, but thank you. Justin. Nothing tonight. Thank you. John. Yes, sir. So I think Mr. Gentilly took his name out of consideration for the moment. I had a conversation with him on Friday, I believe, and it was after seeing the agenda for this meeting where we were reappointing, in this case, it was filled to the conservation board, which I have absolutely no problem with, but it occurred to me that we don't advertise those positions unless if someone just doesn't want to run again. And in my view, we should be, and this is the process that we used in Norfield, and not that it's right, but this is what we did, we would put out an advertise that, you know, a one-year appointment to the conservation committee is open. Please send your letters of interest to the conservation committee at so-and-so address. The conservation committee would then make a recommendation to the select board, and the select board would ceremoniously appoint someone. But at least that way it gets the advertisement out there, and maybe, you know, maybe there's some new people in town, maybe there's, you know, a new person that lives on Berlin Pond that's a forester, or you just never know if you don't advertise and try to get new people on the board. So, and in this case, obviously nothing at all against Bill. And I hope he's reappointed, he does a great job. But I think that we ought to be advertised in these openings, and letting people know that, you know, there's openings in the town that they could, you know, show interest in if they'd like to, and potentially be on one of these boards. Well, we're usually taking, of course, this year is going to be different, but usually at town meeting, we take and go through a list of vacancies. I get that, but usually, you know, at town meeting, there's, you know, 80 to 100 people there. I think, I really think we ought to be advertising where we reached the full 2800. People are busy, you know, not everyone is interested in, you know, maybe sitting through town meeting, maybe some people have to work, but maybe they're interested in the conservation committee or the planning commission. And I just think that, you know, if we want really good quality boards, we should be advertising and, you know, trying to, you know, freshen our blood and get the best people on the boards that we can. No different than the select board or anything else. We're all up for election every so often, and we, you know, it goes out to advertisement and we run again if we want to, or we say no and yeah, whatever. But anyways, that's the way we did at Northfield, and I just thought that, you know, maybe it was a good way to approach things here in Berlin. John, go ahead, Brad. Sorry. Well, the only thing I was thinking is that I got, we'll have to talk to Rosemary, but if we could take and put a list of openings in the ballots, if she's sending those out to every, all the registered voters in town. And I would think with Corinne's emails that she sends out along with Front Porch Forum that we would be able to get that out there. I want to follow up with John's statement, I think that's a great idea to get fresh life, new energy, that always helps. Do, I mean, I don't even know the, is there, is there a set number of members that can be on any of these, the conservation committee or any of that? Because from what I understood, there's been various amounts of different members. And, and to me, that's also confusing. But I think it depends on the board or commission, and Brad, correct me if I'm wrong, but Phil was going to do some research into the conservation committee and hopefully get us an answer in the next week or so, as far as what the number is, if we can't find it. Because the conservation board is going to have a recommendation to the select board, as far as what the number should be. Well, the, the, trying to think, the zoning commission, I, at one, at one time they had five, but at one time I think they had seven. One time the planning commission had seven, they couldn't feel the quorum and they took that down to five. Yeah. Right now, the development review board is five. The public work board is, is five as well. What's good about the development review board, they have the ability to, to use alternates. The planning commission is not allowed, I think it's my charter not allowed to use alternates. And the public's work board, I believe is under the same scenario that, that they can't use alternates. There are openings on all three of those boards. I think what's, what, what it's really from, from a staff standpoint is having one person in charge of doing that. And I would suggest that if, with the new hours to the assistant town clerk, she's, that maybe the select board asked if she could do this task. And because that's what ends up slipping through the clack, the cracks. Okay. Anything else on your round table, John? No, just, just, just so we're all on the same page, though, our, our, what do we want to do? Because I don't want to leave a conservation committee member out there that wants to serve out on the wind, not knowing what our next steps are. Do we want to start advertising for each position, even if it's up for reappointment, or, you know, a two week period? And at the end, or do you want to work on this over the, you know, next two weeks with, or in the course of February and have our new town administrator take this on and develop a process for us to look at, like, how would you, what's the board's wishes here? The, so there's, how many are on the, on the conservation committee? I believe it. We don't, I believe it's fine. I believe it's fine. Yeah. And five, five people. Yeah. Well, I mean, if we're going to advertise for the commissions, I don't see any problem with that as long as we advertise for all of them. Great. They all, they all need, they all need more members or alternates. Agreed. And then the process would be that, that in, in the case of the conservation committee, they would review the applicants and bring a recommendation to the select board or select board approval. It, it probably should be for any of the committees for the existing members to take in, review them. I have no problem with that. Well, I, I'm not sure the wisdom of that because you, if you've got a full board, of course they can always up their numbers to, to seven. They can come before the select board and ask to have their numbers raised. But I don't see anything wrong with that.  I mean, there's always conflicts. I, I guess I would go for, you know, we've got a management plan to write, which is a very labor intensive activity. We've got the research now to do on the vast trail, which is going to be another labor intensive activity. We have Phil, who's an experienced person on the committee. He's part of our historical knowledge of the committee.  Phil is the second longest standing member. He's currently our chair. And now we've got, you know, two big projects coming up besides normal things that we need to do. And in the past month, I have attempted to recruit three people onto the committee to get extra people. And all of them, although they've been interested, they, you know, they've expressed interest in the activity. They expressed interest in the forest. They don't have time to take this on. And my thought is, if anybody is willing and able to do this, we, as a committee, welcome them. You know, especially an existing member with experience. Yeah. So what you're saying is this makes sense to me. You're, you would recommend appointing Phil back to the board. And in the meantime, we could also advertise for openings. And should, should any, anybody come to that board that's willing to put in the time and the energy, you would, you would recommend most likely you can't guarantee anything, but you would recommend that they would also be able to be a member of that board and you'd expand to get the maximum involvement that you could. Yes. I mean, we need people. And it is, like I said, I've tried to recruit three people in the last month unsuccessfully with good experience and good backgrounds. And Phil has a lot of experience with this committee that we need. You know, so to put any, anything in the path of him not being back on the committee is a mistake. That, that is not, and just to be very, very clear, that is not the intention whatsoever. And I'm, I'm happy to reappoint them immediately if that's, if that's what's, you know, that's the desire of the committee. My, my point was that how do you get new fresh blood if you don't advertise the positions when they come up for reappointment? Advertise, we would like more people. That's what I'm, that's, that's why I'm, that's why we're having this discussion. As I said in the beginning, it has absolutely nothing to do with Phil. So in the, in the process of doing this, John, couldn't you, couldn't the board, wouldn't you think, wouldn't the board be able to receive applicants or letters of interest? So to help determine whether or not the select board felt as though maybe they should go from five members to seven members. And I mean, obviously in, in conjunction, but that way you would also eliminate that potential for kind of old blood, so to speak, on the board and get fresh new energy in there sometimes. Three new members currently. So we don't really have an issue with having the old blood. We have an issue not having enough experience. I didn't mean it like that, but if it got to be a stagnant environment or, or what, I'm just, it's not, not implying that that's the case there at all in any way, shape or form. Just like, I think Phil's done a fantastic job and everybody does a fantastic job on there. It was more if I think, you know, Brad was saying, if, you know, when you think about it, if there's five members and, and they're the ones that bring the recommendation, sometimes there could be a bias or unknowingly or just a comfort or, well, maybe you wouldn't take that person on because this person's got a bunch more experience in the town or in that area. So that was my only point was that I think it would be, I think it's great. And I think that if, if letters of interest came to the, the select board and, and we thought there was a tremendous amount of qualified people and not enough positions on a board, then, then we would be able to make a recommendation to these other boards that they maybe change from a five member board to a seven member board or something along those lines based on the amount of enthusiasm or willingness of, of the community to be engaged in that board because we all know the more people, like you said, to handle these tough tasks, the better. That's all I was getting at. And this, this is very, this has very little to do with that, the conservation committee specifically. This has to do with overall, how are we making appointments? So if we had a five member board and we just reappointed, if you wanted to stay on those five members could be on for 50 years and never get off and no one would ever have a chance to get on as a new member. That's what I was trying to point out. It's not an issue. It's not an issue. It's a lot of work, not to pay, it's not an issue. Again, you got to look at it beside bigger than just the conservation committee. I realize that's the board you care about specifically, but we're trying to look at this holistically across all of our committees. And I think all of them are members. Okay. All right. Everybody from all right. Conservation committee to the DRB. Good luck. Good luck with a good, good luck with a quorum though. If you, if your group gets too big as with named members, but we'll take your recommendation on whatever number you want to give us. Well, we don't vote. So it's okay. Okay. Anything else? Oh yeah. So I want to drill out of this. So what you're saying is you're not a real committee. No, I'm saying we don't have the same voting power that this like board has. We're an advisory committee. We don't vote. Do you hold public meetings? We're advisory. We don't need a quorum to vote. We don't vote. And we don't really need to appoint anyone. Okay. I'm good. Okay. I have nothing. So Tom, is there executive session tonight? No, sir. Okay. I want to entertain a motion to adjourn. Motion to adjourn. Second. All in favor. Aye. Aye. Flo, was that an aye? Yes. You have three ayes there, Mr. Chairman. We are adjourned.


This lecture is part of Berkeley Math 115, an introductory undergraduate course on number theory and will be about Dirichlet's theorem. So, Dirichlet's theorem says that if A and B are co-prime positive integers, then there are infinitely many prime to the form A n plus B for n greater than or equal to one. For example, if we take A equals 10, B equals one, this means there are infinitely many prime to the last digit one, and so you can see there's sort of 11, 31, 41, and so on. So, we've proved this in various special cases before. So, in the case B equals one and A equals P a prime, we showed there were infinitely many numbers of the form nP plus one by considering the polynomial x to the p minus one over x minus one, which is x to the p minus one plus x to the p minus two, and so on, plus one. There's a variation of this where you can show there are infinitely many prime to the form one modulo A, even when A is not prime, by using something called a cyclotomic polynomial, which is a polynomial whose roots are primitive and ath roots of unity. So, this is the cyclotomic polynomial for a prime P. And we also did various other cases that were quite easy. For instance, for n, for a plus three, so for n plus three and three n plus two, followed by an easy variation of Euclid's proof, that there are infinitely many primes. And I guess, of course, the case A equals one is also easy by Euclid's proof. We gave some other proofs for progressions like four n plus one or three n plus one by using other polynomials. For four n plus one, we consider the polynomial x squared plus one, for example. And for eight n plus three and eight n plus seven, we gave a proof using properties of the quadratic residue symbols, two P and minus two P. And we used polynomials x squared plus or minus two. So, there are quite a few cases when we can give sort of reasonably easy proofs of Dirichlet's theorem. The problem is none of these methods seem to generalize to the general case. For example, these methods mostly seem to work if all elements of the group z modulo nz star have ordered two. And this only works if n is a divisor of 24. So, we get one, two, three, four, six, eight, and for that matter, 12 and 24. So, for these numbers, we expect to find a reasonably easy proof, but in general, this just doesn't work. So, what Dirichlet did was he used Euler's proof that there are infinitely many primes. So, we just quickly recall that if you write zeta of s, which is one over one to the s plus one over two to the s and so on. And Euler showed this could be written as an infinite product, product over one over one minus P to minus s. Well, this is a product over all primes. And Euler simply noted that zeta of s is infinite at s equals one. And at s equals one, this product becomes the product over all primes of one over one minus P to minus one, which is finite unless there are an infinite number of primes. And in an earlier lecture, we showed there was a variation of this. You could show there are infinitely many primes that are one or three mod four by using the series L of s, which is one over one to the s minus one over three to the s plus one over five to the s and so on. And this also is an Euler product. It's the product over P of one minus one over one minus chi of P times P to minus s, where chi of P is equal to plus one if P is congruent to one mod four and minus one if P is congruent to three modulo four. Now, the idea is that this series is none zero and finite at s equals one. I mean, it's just an alternating series. And you can see that it's none zero because the sum of these two terms is positive and the sum of the next two terms is positive and so on. So that means this product is actually finite. And what this product is, it's a product over P congruent to one mod four of one over one minus P to minus s and the product over P congruent to three mod four of one over one plus P to minus s. And the fact that this is finite sort of means that there are roughly the same number of primes that are one mod four and three mod four because if there were too many primes that were one mod four, this half would dominate and the product would be infinite and if there were too many primes that were three mod four, this product would dominate and the result would be zero. So the fact that this is finite and none zero at s equals one sort of means that in some vague sense, the primes that are one mod four and three mod four kind of balance out. Of course, we need to make this balancing out idea a bit more precise. I'm just giving a rough idea. And since we know that there are infinitely many primes altogether, this means there must be an infinite number that are one mod four and an infinite number that are three mod four and Dirichlet's proof basically consists of taking this idea for primes modulo four and generalizing it to arbitrary congruences. So let's just summarize. Let's just give an overview of Dirichlet's proof. We can say that there are essentially three steps. The first step is we need to define analogs of chi of n. This was the one that was plus one for one mod four and minus one for three mod four. And we need to define the analog of LS. So we know that for modulo four and we need to define them for other things. The next thing you notice is that a key step in the proof was to show this function was none zero at s equals one. And from that, we were able to deduce the Dirichlet's them. So the second step is to show that so L chi of one is none zero. So L chi is going to be the generalization of L of s. We're going to generalize chi to other so-called Dirichlet characters and define an L series for them. And a key step is to show these functions don't vanish at one. And the third step is to show that L chi of one being none zero implies Dirichlet's theorem. And in some sense, part three is reasonably straightforward. Once you've had this idea of examining these so-called Dirichlet's theorem L of chi it's not too difficult to deduce Dirichlet's theorem from the non-vanishing. The really tricky part of the proof is to show that these functions are all non-vanishing at the point one. So this is rather strange. It's sort of deducing this number theoretical fact about prime numbers from this real analytic fact that certain infinite series converged to a non-zero number. You remember that when we were discussing the prime number theorem, we showed that the prime number theorem depended on the fact that zeta of s is non-zero whenever the real part of s is equal to one. So in both cases, we see that the behavior of prime numbers is controlled by whether certain Dirichlet series vanish. So for the prime number theorem we want the Riemann zeta function to be non-zero on a certain line. For Dirichlet's theorem, we want these L series to be non-zero at s equals one. And this is very much a theme in analytic number theory. The prime seem to be kind of controlled by zeros of certain Dirichlet series. Anyway, so let's define what Dirichlet characters are. So first of all, we look at this character chi, which took values one, zero, minus one, zero, one, zero, minus one, zero and so on. And we're trying to think what properties of this did we need. The first one is that chi, this chi is periodic mod four. So we now want to define a Dirichlet character, modulo n. So the first property we're going to say is that chi of an m plus big n is equal to chi of m. So it is period n for some number n. Secondly, we notice that it vanishes, that this chi vanishes whenever its argument is even. And the generalization of that is going to be that chi of m is equal to naught whenever m and n are co-prime. And the third property is that this is actually multiplicative. So chi of m, n is equal to chi of m, chi of n. So this was used in writing down the Euler product for the corresponding Dirichlet series. And these are more or less the definition of a Dirichlet character. We should also add an extra condition to stop it being completely trivial. I mean, it could just be zero everywhere, which is not terribly interesting.  Well, there's an alternative way of thinking about this. Chi is a homomorphism of groups from z modulo nz star to the non-zero complex numbers under multiplication. And a homomorphism is just condition three. Here it says it preserves multiplication and conditions one and two. Well, condition one says that it's defined on z modulo nz and condition two says you can ignore its values on the elements of z modulo nz that are not co-prime to n. So a sort of modern definition of a Dirichlet character would just be this one here. Dirichlet's original definition was like this because the concept of group hadn't actually been defined at the time, so we needed to do everything by hand, so to speak.  And it's just chi of one over one to the s plus chi of two over two to the s plus chi of three over three to the s and so on. So we'll be using this Dirichlet series quite a lot. So what we should do now is just go through some examples in order to see what these things look like. So let's start with n equals one. Then the only Dirichlet character is chi of n equals one for all n because chi of one equals one and it is period one. So the l series we get, l chi of s is just one over one to the s plus one over two to the s and so on, which is just the usual Riemann zeta function. So that wasn't terribly interesting. And of course it has the usual Euler product, product over p of one minus p to the minus s. So now let's try n equals two. So in this case, c modulo two z star just as one element. So there's only one possible that chi, which goes one, zero, one, zero, one, zero and so on. So chi is one for n odd and zero for n even. So we get the series one over one to the s plus one over three to the s plus one over five to the s and so on. But this isn't really a new function because what we notice is that it's equal to a product over p odd of one over one minus p to the minus s. So in other words, it's just equal to one minus two to the minus s times zeta of s. So it's not really a new function. It just differs from the Riemann zeta function by this rather elementary and uninteresting factor. So next we try n equals three. And here's z modulo three z star has two elements, one and two. And now there are two possible values of chi. First of all, it could just be one everywhere, but it could also be minus one on the character two. So we get two L series. One is one over one to the s plus one over two to the s plus one over four to the s plus one over five to the s and so on. And the other one is one over one to the s minus one over two to the s plus one over four to the s minus one over five to the s and so on. And this one again turns out to be more or less the same as the Riemann zeta function. It's just one minus three to the minus s times zeta s. So it's not terribly interesting. This one on the other hand is sort of a new L series. Now let's take a look at n equals four. So we get z over four z star. And this again has two elements, which are one and three mod four. So there are two possible Dirichlet characters. The Dirichlet character can be one everywhere or it can be minus one on three. And if it's one everywhere, we get one over one to the s plus one over three to the s plus one over five to the s and so on. And you may notice this is exactly the same as a character we had for n equals two, which was in turn more or less the Riemann zeta function. So this isn't new. It's just one minus two to the minus s times zeta s. And for this one, we get the L series we mentioned at the beginning, which is one over one to the s minus one over three to the s plus one over five to the s and so on. And we should notice that this converges for s equals one and is none zero because as we said, we can see it's none zero because these two terms have a positive sum, the next two terms have a positive sum and so on. And while we're about it, we can notice that this series is none zero because these two have a positive sum, these two have a positive sum, the next two have a positive sum and so on. So this is none zero at s equals one. So what about n equals five? Well, this starts to get a little bit more complicated. So there are four possible elements of z modulo five z star. And how do we find characters? Well, what we notice is that we can write these as two to the zero, two to the one, two to the two and two to the three. And we notice that two to the four is equal to one. So chi of two to the power of four must actually be equal to one. So chi of two must be one i minus one or minus i because these are the only complex numbers whose fourth power is one. And once we figure out what chi of two is, everything else is determined. So we get four possible characters. The first one is one everywhere except at five. The next one goes one i minus one minus i, the next one goes one minus one minus one one and the next one goes one minus i, i minus one. Now you notice for the first time we're actually getting some non real complex values of these characters. And let's look at the L series. Well, this one as usual is one over one to the s plus one over two to the s plus one over three to the s plus one over four to the s. And then we miss out one over five to the s and we get plus one over six to the s. And as before, this is just one minus five to the minus s times zeta of s. So it's not really new. These ones, however, are all new characters. So let's look at this one first. It's a little bit easier. We get one over one to the s minus one over two to the s minus one over three to the s plus one over four to the s. And then we get minus one over six to the s. So plus one over six to the s minus one over seven to the s minus one over eight to the s plus one over nine to the s and so on. And now let's see that this does actually converge to something that's none zero. So let's arrange the terms in pairs first. So we see this sum is positive and this sum is negative and this sum is positive and this sum is negative and so on. But all of these, but these terms I put in circles are getting smaller and smaller and absolute values. So what we really have is an alternating series. We have something minus something smaller plus something smaller still minus something yet smaller. And from this, you can see that it converges is so it's finite and none zero. And then there are the two other complex ones to look at. So let's just take a quick look at those and see what happens. So one of them would be one over one to the s plus i over two to the s minus i over three to the s minus one over four to the s and so on. And now we can see that this is none zero at s equals one and that's fairly easy because we can just look at the real part. So the real part is going to be one over one to the s minus one over four to the s plus one over six to the s minus one over nine to the s and so on. And this is alternating with decreasing terms. So you can easily see that it's none zero and finite at s equals one. The other one is just one over one to the s minus i over two to the s plus i over three to the s minus one over four to the s. And you notice immediately, this is just the complex conjugate of this function here. So anything we say about this function, we can say something similar about that function. So that's quite common for these Dirichlet L series and characters to come in complex conjugate pairs. Now let's take a look at n equals six and here the numbers modulo six that are co-prime to six are just one on five with five squared congruent to one. So there are only two possible characters, one one and one minus one. The first one gives us one over one to the s plus one over five to the s plus one over seven to the s plus one over 11 to the s and so on. And what you notice is that this is just a product overall primes other than two and three of one over one minus p to the minus s which is equal one minus two to minus s times one minus three to minus s times zeta of s. So you see it's again essentially the same as zeta of s although now we've got two different primes we need to sort of take out of the infinite product for zeta of s. And the other one is one over one to the s minus one over five to the s plus one over seven to the s minus one over 11 to the s. And this is more or less one of the functions we've had before. So you remember for p equals three we had this function one over one to the s minus one over two to the s plus one over four to the s minus one over five to the s and so on. And what you notice is that these two are really the same except I've multiplied to get from this one to this one I have to multiply by one over one minus two to the minus s. So it really differs from the function we had for n equals three by just some sort of elementary Euler factor. So we haven't really got anything. Sorry, that should be a plus sign. We haven't really got a new function. Both of the functions for n equals six are the same as functions we had for n equals one and n equals three, except we have an extra elementary Euler factor. So for six, we get nothing new. And for seven, the numbers modulo seven are one, two, three, four, five and six. And now we're going to repeat the argument we had for n equals five, except we may as well do it for all primes p. So the key point is if we look at the numbers one, two, up to p minus one, mod p, we see that there's a primitive root v. So for seven, we can take a primitive root to be three, five or six. It doesn't really matter which. And we can think of these numbers as being g to the zero, g to the one, g to the two, up to g to the p minus one and g to the p equals one. So chi of p is a p-th root of unity and there are p of these. So we get exactly p characters. So the characters are as follows. We have g to the zero, g to the one, up to g to the p minus one. And we could have a character that's one on them or a character that's one and epsilon and epsilon squared and epsilon cubed or it could be one, epsilon squared, epsilon four and so on. Here, epsilon is going to be e to the two pi i over p. So epsilon to the p is the one. One epsilon cubed, epsilon to the six and so on, all the way up to one, epsilon to the p minus one, epsilon to the two p minus two and so on. So we have p characters for any prime p and the first character gives us something rather boring. It's as we saw with two, three and five, what it does is it gives us one minus p to the minus s times zeta of s as the L series. So we don't get anything new. The others all give us new L series except these come in complex conjugate pairs as we saw for p equals five. There are various pairs and maybe there's, there might be one character in the middle that's actually real. In fact, yes, there's always one character that's real. So that does all primes. Now let's look at n equals eight. So here we have one, three, five and seven as the elements of z modulo eight z star. And to all previous examples, we've been able to work out the characters just by picking a primitive root and saying that all elements of powers of that primitive root and from that it was easy to find the characters. But we can't do that for n equals eight because eight doesn't have a primitive root. So we have everything squared is equal to one. And this means chi of three squared equals chi of five squared equals chi of seven squared equals one. So chi of n is always equal to plus or minus one unless n is even in which case it's zero. And we also have chi of seven is equal to three times five mod eight. So once we've chosen chi of three and five, this determines chi of seven. So we see there are four possibilities. One, one, one, one, one, one, minus one, minus one, one, minus one, one, minus one, and one, minus one, minus one, one. So let's take a look at these. Well, first of all, this one is the one we've already had it's one over one to the s plus one over three to the s and so on, which is one minus two to minus s times zeta of s. So that wasn't very interesting. This one we've actually also had before it's just the one over one to the s minus one over three to the s plus one over five to the s which we had modulo when we were working modulo four. These two, however, are new. So we get one over one to the s plus one over three to the s minus one over five to the s minus one over seven to the s plus one over nine to the s. And this one we get one over one to the s minus one over three to the s minus one over five to the s plus one over seven to the s and so on. And we should think about these a little bit and work out whether or not they're zero. And if you think about it, it's quite easy to see they're both non zero. For instance, this one has two positive terms then two negative terms then two positive terms then two negative terms. But these two are bigger than these two. So the sum of these four is definitely positive. And the sum of these four is also definitely positive for the same reason and so on. These ones you have to think about a little bit more carefully. And we can use the same argument we used five. So some of these two is positive and some of these two is negative but less in absolute value than that one. And the next two is some is positive and the next two, the sum is negative but that's less than that in absolute value and that's less than that in absolute value.  So the sum is non zero and finite. So we've checked that for n equals eight all the Dirichlet series are non zero and finite. And now let's look at n equals 12 and 12, there are four numbers one, five, seven and 11 modulo 12. And we could do this like the case n equals eight because you can see that all of these have square equal to one but what I want to do is to use a more general argument. So we're going to write n is equal to three times four and we notice that z over 12 z star is commutant as z over three z star times z over two z, so it's a z over four z star. And now if you've got any homomorphism from this to the complex numbers, you can see it gives you a homomorphism from z modulo three z to the complex numbers and a homomorphism from z modulo four z to the complex numbers. And conversely, if you've got a homomorphism from each of these groups, the complex numbers, you get a homomorphism from this to the complex numbers. So we can now see directly how many Dirichlet characters there are. This is two characters and we saw z modulo four z also has two characters. So we can work out the characters of z modulo 12 z. We just get two times two characters and we can do this whenever n is a product of different prime powers. If you want to write this out explicitly for n equals 12, we can see this. So n equals 12, we have one, five, seven, 11 and the possible Dirichlet characters go one, one, one, one or one, minus one, one, minus one or one, one, minus one, minus one or one. Here's the other one, one, minus one, minus one. So what you notice is that this one is really character modulo three because it only depends on the value of this modulo three and that one's also a character modulo three. This one is a character mod four. So, sorry, that one. So we've got two characters mod three and two characters modulo four and all the characters mod 12 obtained by multiplying one of these characters mod three by one of the characters modulo four. So this one, for example, is given by taking the non-trivial character mod three and multiplying it by the non-trivial character modulo four. So we can now figure out how many Dirichlet characters there are mod n. So how many characters are there mod n? So first of all, if n has a primitive root, then g, then the elements of z modulo nz star are just one g, g squared, up to g to the phi of n. And the Dirichlet characters are given by mapping g to epsilon to the k where epsilon is a primitive nth root of one. So epsilon to the n is equal to one. So we see that there are exactly phi of n characters. In particular, this works whenever n is a power of a prime for p odd. For n power of two, we can do something similar because we know that z modulo nz star is then can be written as a product of two groups, the group plus or minus one times the powers of five. And now we can work out the Dirichlet characters modulo n in a rather similar way because there are two choices for the image of one, this can go to plus or minus one, and there are two to the k minus two choices for the image of five. So again, we get phi of n characters. And finally, if n is equal to p one to the n one times p two to the n two, and so we apply the Chinese remainder theorem. So z modulo nz star is z modulo p one, the n one, z star times and so on. And the characters of z modulo nz star can be written as a character of the first group times a character of the second group and so on by the Chinese remainder theorem. So the number of characters of z modulo nz star nz star is equal to the number of characters of z modulo p one, z star, which as we said is phi of p one to the n one times phi of p two to the n two and so on. And Euler's phi function is just multiplicative. So this is phi of n. So the number of characters of z modulo nz star is equal to phi of n, which is equal to the order of z over nz star. So this is a very basic fact that we're going to use next lecture when discussing further properties of the characters. So the next three lectures, the next lecture will be more properties of characters and L series. The lecture after that will be how to deduce Dirichlet's theorem from non-vanishing of the Dirichlet L series. And the final lecture will be on how you show Dirichlet L series on non-zero.