 Thank you very much. I thank the organizers for inviting me to speak here. So let's get right down to business. The Riemann Zeta function. In a celebrated paper written in 1859. Riemann showed that the series summation one over NBS. Defines an analytic function, the region realize bigger than one can be extended analytically to the entire complex plane, with only a simple polar s equals one with residue one. Moreover, it satisfies this remarkable functional equation. One half SS minus one pilot minus two gamma s or two Zeta s is equal to CF one minus s. So half will appear will become important later as we talk in this lecture. In the same paper he noted the existence of the Euler product Zeta SS product over prime numbers one minus one of the previous inverse. He gives us a link between the properties of the Zeta function and the distribution of prime numbers. So to underline this, he gave a heuristic derivation of the explicit formula for Pi of X, the number of primes of X in terms of the complex series of Riemann Zeta function so an amazing paper, full of ideas, perhaps not fully exhausted even as of now. And in that paper, motivated by the symmetry of the functional equation, along with some preliminary computations. He stated was now called the rebutt hypothesis that all complex zeros and Zeta s lie on the line realize equal to half. This hypothesis along with various generalizations is still an unsolved problem, however, there have been reformulations that link it to probability theory and so I want to talk a little bit about those links in this in this lecture. So we'll go deeper into this connection first review the possible summation formula and the derivation function equation. So given a C infinity function f we define its free a transform as f hat of t, f of integral of f of x C to minus two pi xt dx. And then the possible from the summation formula is simply summation f of n is the same as summation f had event. So this is a particular definition of the three transform. Now, if we can apply this possible summation formula to this function f of t, either the minus pi a plus t over root x pole squared, then a simple computation using your favorite method shows that f hat of t is root x equal to pi at root x either minus pi t squared x, essentially coming from the fact that either minus pi t squared is its own full year transfer. And then you apply the possible summation formula long before you get this beautiful identity summation either minus pi a plus n over root x full squared is equal to square of x times either minus pi and squared x to pi I am sure all of you have seen this before. And in particular when a equals zero you get what's called Jacobi theta functional equation which is summation either minus pi and squared over x is equal to root x either minus pi and squared x. This is basically a Riemann starting point he notices that this functional equation of the theta function will imply a functional equation of the Riemann's data function and thanks to, I believe it's hecke or others, we know or hamburger or something I think hamburger maybe it goes backwards as well. What theta of t is equal to either minus pi and squared t. And so the above identity translates to the functional equation, theta of one over x equals squared of x, theta of x. Now, this is a modular form of weight one half and the ambiguity of the square root side has always been an important feature, and it is remarkable that in the theory of modular forms and this understanding of this, this transformation formula didn't appear until she was theory of having to wait multiple forms. So the functional equation for the data function begins with Riemann's paper where he notices that if you take the gamma function. t equals zero infinity minus t to the s minus one d t and change variables, but t equals n squared x or something and then you get gamma that's over to enter the minus s. And as an integral involving either minus squared x x to the s over to the x over x, changing x to pi x in the integral and summing over and we end up getting that pi the minus s over to gamma sort of to z to s is the transform of the Jacobi data function. So this is this is straight out of Riemann's paper. The function of brackets is not quite the data function as you will know the data function from minus infinity to infinity. So you have to modify that so the modification is done as the obvious sort of way to also put w x to be half of the data function. And we see that the functional equation for the data function translates into a functional equation of the w function and injecting that into our integral. And we see that the integral can be split into two parts. And once you split the integral into two parts and the first integral change x to one over x, so that the integral then becomes an integral from one to infinity again. So for example, you get this transformation. And finally that this equals x minus s over two times these residual terms, and then those residual terms contribute one over s and one over s minus one. And then long before you get into one to infinity x to the one minus two w of x, the x over x so putting everything together. And combining these calculations you get part of the minus that's over to gamma server to zeta s equals one over SS minus one integral one to infinity w of x x to the sort of two plus x to the one minus that sort of do the x or x. So w x is this truncated jupy function. So you have this function satisfies the functional equation zeta, sorry, c of s equals c of one minus s, and it extends to an entire function because this integral, this integral here is an entire function because now w of x remember starting from one to one is exponential decay, and therefore this integral makes sense for all values of s. So that's the proof of the functional equation to beautiful derivation. And the reason for putting a half in front of the c function will become apparent as we go on with the talk. So the functions g of y and h of y. If we change the variables instead of looking at theta y theta y squared and call that g of y. Then the functional equation for theta translates to g of one over y equals y g of what you remove the ambiguity the spirit of science, what I suppose this is one of the motivations for making this change of variable. You can rewrite your CFS as to CFS SS minus one integral zero infinity g of y minus one y the SD y over y. And if you set h of y to be d by d y y squared d by d by g of y. And then you get a differential equation involving g of y to y g prime of y plus y squared g double prime of y. And noting that g prime of y can be written as a summation pi and squared y in the month plan squared by squared and g double prime of y can also be similarly written you have these terms. So the h of y is actually four y squared times summation and one from one to infinity to pi squared and to the four y squared minus three pi pi and squared minus pi and squared y squared. So why am I going through all this, it's because this is going to be important h of y and g of y turn out to be non negative functions. So CFS is a melon transform h of y so h of y satisfies the same functional equation as g of y and it's not obvious of course you have to check it. And just leave the details because the stock is being taped, you can see these slides at your leisure, but basically, you do the obvious differentiations and check routine verification that h of one of y is actually why times h of y. The formula for CFS then becomes to CFS integral zero infinity h of y while the s dy or y. In other words, I've expressed CFS as a melon transform of h of y, but h of y is this funny modified data function. And two pi squared into the four y squared is bigger than three pi n squared for why bigger than equal to one. We see that h of y is positive in the region why bigger than equal to one. But because h of y satisfies this functional equation. Otherwise also h of one over y is why h of y, therefore it satisfies positivity condition for why positive as well. So, in other words, the bottom line is that took CFS is the melon transform of a positive function h of y is not a positive function. And this is all due to polio. And I'll talk about polio in a second. In other words, CFS is the melon transform of a non negative function h of y. That's the, that's the starting point for much of the discussion of the connection between probability theory and we want to. If we, we've proved that took CFS is the interval zero to infinity h of y, why the s dy or y, the observation will not be used to move into a probabilistic view of the data function, checking what c of one is in the usual way away. You find that it's equal to half and that explains the two, two times C of zero is the same as two times C of one is equal to one. In other words, we have the integral zero infinity h of y or y dy is equal to integral zero infinity h of y dy and it's equal to one. In other words, h of y being non negative. And the integral being equal to one allows you to interpret h of y as a probability density function. This is an amazing, amazing idea. As I said, it's all due to polio. Therefore, h of y can be viewed as a density function of a probability distribution on zero infinity and these intervals tell you that the mean value of this distribution is one. This is also deduced from the same formula because of the positivity that CFS has no positive real zeros. This is going to show up later on in the discussion, but out of this, I mean there are other ways of showing that the data remand data function has no positive zeros, but this is one way of doing it and this becomes important. When I want to discuss current developments. So the probability density h of y. So far we've shown that to CFS is the melon transform h of y h of y is positive for positive. So in general, if you take a random variable with this distribution h of y, the functional equation then translates as the expected value of f of one over y equals the expected value of y times f of y. And in particular, if we put y to be y v s is the as the random variable one can view to CFS as the expected value of the random variable y p s. All of this, as I said is due to polio 1926. And certainly it was part of the Hilbert polio dream of finding a Hermitian operator whose eigenvalues are equal to i times Roman as I have a rule once through the zeros of CFS. Notice, by the way, the trivial zeros of the remote data function do not appear anymore in CFS, they've been eliminated so CFS the zeros of CFS are the non trivial zeros of the remote data function. And polio devoted a substantial portion of his research to the study of Fourier transforms of functions, all of whose real zeros are real. So in one paper he shows that if we just take the first term of the free expansion of h of that h of y the first answer it's not free expansion that data function data function like expansion right. It's either minus pi and squared t with certain coefficient, if we just take the first coefficient first term. He actually wrote a paper in which he analyzed the zeros of the melon transform of that thing and showed that all the zeros are real corresponding to what you probably expect for the C function as well. Now, in 1995 97, Jean-Jean Lee derived an elegant criterion for the truth of the Riemann hypothesis. Define these numbers lambdas of N summation row one minus one over the end, where the sum is over non trivial zeros of the Riemann data function. Then Lee's criterion is that the Riemann hypothesis is true if and only if all the lambda ends are not negative. This is a very famous paper of Jean-Jean Lee. We want this to hold for all natural numbers in. Now it's not difficult to see that the lambdas ends can be rewritten as one over n minus one factorial the nth derivative of s to the n minus one times log of C of s evaluated that is equal. We're taking the Laurent expansion of s to the n minus one logs or whatever this and then calculating the coefficient. So this is one interpretation of the lambdas of N's in terms of the law of the C function. Now, given our probabilistic interpretation of CFS as the expected value of the random variable Y vs. The Lee constants now also have a probabilistic interpretation in terms of cumulants, cumulants between now review. So let me give you a quick crash course on what cumulants of a probability distribution are. Given a random variable X, the moment generating function is the expected value of the tx and the cumulant generating function is log of the expected value of the tx. And one can relate the moments and to the cumulants as follows. So if you write the expected value of the tx easy to see that the coefficients are the moments of the distribution. If you write the series as one plus s of t, then s of t converges let's say in a sufficiently small neighborhood of the zero and you can take a log of this thing and make an expansion. And then we see from this that the cumulants, which are the coefficient the coefficients of the log of the tx. So you need to look at the coefficient of the t of the K that that's the case. So you should call it K, the coefficient of t to the end in this expansion will caught the end cumulant. And so you can write down formulas for the cumulants in terms of the moments of the original problem distribution. So K1 is the mean value of X and one is the first moment that's the expected value of the mean value. And K2 is the variance, the expected value of X minus the mean square. And that's always positive isn't it expected value of non negative functions always not negative density function therefore it's not negative K2 will always be positive for any random variable X. So these two comments that K1 and K2 are related in this fashion will become important as we as we proceed. So the relation between the cumulants and the coefficients so one can in fact even explicit relation between the moments and cumulants as follows. So you put f of t equal to either tx and then you just look at the log of that thing that's how you get the cumulants to the cumulant generating function just a lot of that. And we differentiate it so that we will get to handle on the things and multiplying true by t times f of t, comparing the coefficient to the end on both sides. And a beautiful formula, that the end moment of the original distribution is recursively determined in terms of the previous moments and the previous cumulants. And in other words, the end moment can be written as the summation and minus one choose j m sub j times case of n minus j. So that is a random variable with density function h of y as we said the beginning. Let's put L equals log of one over why. Then we go back to our probabilistic interpretation of the women's data function took CFS as the expected value by the yes, which is also by the functional equation the expected value of why the one minus s. And when we write why is, you know, L equals log of one of why we can write the expected value of the s minus one times L. And we can expand this series in terms of s minus one, and we get these cumulants. And so this gives us a probabilistic view of Lee's criterion. We've already noted that the Lee criterion numbers are given by this is this formula and again, and using the Leibniz rule, we can rewrite this thing. The Leibniz rule, meaning if you want to compute the answer or the product of two functions, then you have an expression of binomial expression like this. And then lo and behold, you get a nice clean formula of the lambdas of ends in terms of the cumulants of the distribution attached to the women's data function. So, so the cumulants case of N are just the nth row of log CFS evaluated at 61. So if all the cumulants now were positive. Then you will have the Riemann hypothesis. So now this is not the Riemann hypothesis Riemann hypothesis factorial. Quite a bit, if they were all positive I think we'll have much more. Unfortunately the case of ends are not always possible case of case of threes negative and case of four is also negative but as you can see they're not that much away from zero. So it's an interesting phenomenon. This is the source of this discussion is all remarkable paper by the end with Pittman and your that appeared about 20 years ago and built in BMS. So for those who are interested. I recommend this paper it's a very well written paper. The relativity of lambda one and lambda two are easy to see we already know the connection between moments and cumulants. In the case of the zeta function, the lambda one coefficient, you know, remember boils down to summation one of a row, and summation one of a row you pair up row with row bar and then you end up getting to lambda one is to turn real part of row over. So it's easy to see the firstly coefficient is non negative in this case. And for most all zeta functions you can you can probably do this as for K2 remember that this K2 was related to the variance, and therefore it's always positive. And therefore, lambda one and lambda two are positive for the remunzata function. So the negativity of some of the higher cumulants is disappointing. However, all is not lost because we can apply a 1999 variation of these criteria and do the barbarian legarious, which offers some hope for this probabilistic approach. So before we abandon the, you know, the cumulant research on the cumulants and their growth rates and how bad they are. We are encouraged by the paper by the various, which I will discuss right now. So variants of these criteria and so Bambirian legarious gave an axiomatic treatment of these paper that's quite general. Let S be a multi set of complex numbers roll, such that zero and one are not in set. And if row is in the set and one minus row is also in the set and row bar is in set with the same multiplicity so sounds like a functional equation here. So these data function or zeta functions of number fields or whatever you would like. And then we just put some convergence condition. And this is all satisfied for any function in the Selber class, all these data functions that we've ever seen in life, I suppose, at least in my part of the world are all of order one, but that's a different animal altogether. But in the case of the classical L functions we have order one functions and this number three is always satisfied. And then the theorem of Bambirian legarious is that the following conditions are equivalent in real part of every element in the set is a half is equivalent to the Lee criterion is equivalent to this weaker criterion that all these numbers are bigger minus C of epsilon, E of the epsilon in. So you see this this criterion is amazing. Because it really allows us allows you quite a bit of leeway. They could be negative that the cumulants could be negative but as long as the, the growth is under control of some sort I mean you have to still do those binomial coefficients but essentially, that's the philosophy. So the now here, the week constants are interesting for other reasons forget the remind boxes they're interesting for other reasons in their own right and the arithmetic nature of lee constants is perhaps to be recollected, the cumulants are related to the stelchus constants, gammas of n which are defined as the coefficient of the wrong expansion of both Riemann's data function at s equals one so these are the Taylor coefficients or moron coefficients of zeta s at s equals one. Gamma zero is the familiar Euler constant one use gammas of ends as a generalizations of this in fact will show that gammas of ends are limited given by these limits that basically minus one to the end or in factorial times these limits generalized Euler constants. And these numbers by the way are interesting in their own right and perhaps be studied for other reasons as well. And this was first proved by child and brings in 1955, but the gammas events are given by this nice neat little formula. And as with all child as papers, they contain beautiful ideas, but are never set up in general. So, for youngsters who are listening. Maybe it's a good idea just to study some of the child's papers. It would be useful here to have a general theorem, with a view to applying it to other data function so here's the general theorem. Suppose that you have a nearest to the series of s equals mission and yes. An analytic for realize bigger than one with only a simple pole that's equal to one with residue one assume that the sum of Tory function a summation and is x plus E of x with E of x, satisfying a modest error term of x over log to be a x for any a then you look at the count expansion of f of s of s one, and you can write down a C sub K to be minus one of the key of the key factorial times this expression. I rubbed on that to the K one you may think it's always not zero. It is if K is bigger than one case equals zero it's not. It's one. So, I prefer to write it like that but anyway this is essentially a beautiful paper and minor modifications in child's paper leads to this I say this because we would like to apply it for Zeta prime over Zeta two. And in the, in the Bumbieri legarious paper they derive a similar cold formulas using base explicit formula there's really no need for that here. So we're more interested in the generalized filters constants as been given by the wrong expansion of Zeta s, Zeta prime of S over Zeta that's about as equals one plus you have these coefficients of minus Zeta prime of Zeta, as being the still just constant and the interest in these constants is that we can follow in Bumbieri legarious who use the ways explicit formula to derive the following formula for lambdas of ends they split it up into four parts. The first part's nice second parts nice third parts nice and it's the fourth part that's kind of interesting. So S one of N is summation and choose J minus one of J one minus one over to the J Zeta J. And S two of N is this expression with the steltas constants. Yep. So, so this is what we have. So since they showed that leaves criteria and can be considerably weakened to proving that if for any epsilon positive as a constant C of epsilon positive six of lambda events are always bigger than minus C of epsilon to the end. The epsilon M will be interested in the growth of the terms two terms here on the right hand side. Interestingly, S one of N, which is the term here can be analyzed very easily. It's almost like first year calculus. If you just do the obvious shoving the series for Zeta J and interchange and analyze it and long behold you get a very nice series and you have to analyze. Does this converge and the answer is yes, and then you apply the integral test to estimate it. And so you end up getting S one of N turns out to be all of N log N. And Mark Coffey, who was actually physicist, who's been playing with these terms for a long time. Sadly, I learned that he died about two years ago. I hope it's not due to COVID, but he certainly was not very old. And he's been interested in in this fun these functions for a long time, and he proved this remarkable formula that S one of N is asymptotic to one half N log N. Plus all these very nice terms so I told you to show all of N log N is first year calculus. But if you do a little bit more careful analysis you can actually get this as a product formula. So in other words, if we're trying to show lambda sub ends are bigger than minus E the epsilon M, you know exponential growth, but negative exponential growth, you may as well ignore the first three terms and focus your attention on S two. And S two contains only the skelch's constants. So this can be now I told you this can be derived without using please explicit formula. I gave a course on probability and number theory last term at the fields Academy, and I actually proved that without the explicit formula. So the study of S two of N. And show it's not too negative that's what we need to do. And these are the numbers and what I find fascinating about the very assertion is that you get the remote boxes by knowing what the Riemann's data function is doing at s equals one. That determines the entire remote boxes, you know, it was amazing. So one can actually prove that the eight as they're not positive unfortunately actually oscillate in science. And this again, I've seen some papers, they're proving it using rather complicated methods. It can be deduced via power series analog of Landau's theorem for Dirichlet series called Prince Heim's theorem. So, many of you may be familiar with Landau's theorem about Dirichlet series with non negative coefficients. Prince Heim's theorem is a power series version of that was proved in 1894 well before Landau's theorem. So the statement is that if zero is less than r is less than infinity if you have power series summation a and z to the n with radius of convergence capital R, and they're all non negative. And the proof is identical actually proof that Landau's theorem proof of Prince Heim's theorem is actually identical. So I'm not going to go through it, of course, but now if you apply this in your case of the Zeta prime of Zeta, and ask yourself, are they a fixed sign after a while. Then you analyze FFS truncated and see the right hand side, let's say it's a fixed sign coefficients, and then applying Prince Heim's theorem tells you that it has to have a real singularity, but there is no real singularity, because Zeta prime of Zeta has no real zero, except from the trivial zeros. And there's no positive real singularity so that's the proof that's the proof of Prince Heim's theorem allows you to show that these ades oscillate in sign. Now, further study of the a to j's from our generalization of Charles Briggs theorem we have that the a to j's can be given by this formula as I point this out simply because you know very various paper gets this formula which is a beautiful form via the base explicit form. So, the lambda sub m is the usual one mongrel function. This expression is very unwieldy, we will try to offer their expression for real linear polynomials. The psi of x is summation lambda m, and less than x, and we can write minus C prime over C of s as the melon transform of psi of x, and doing the obvious separating up the pole, and writing the integral psi of x or minus x over x the s plus one allows you to expand the integral into a expansion and get a formula for all these. Ades days I mean these steltos constants steltos constants can actually written down in terms of the error function and delta capital delta x is minus psi of x minus x. And so if we let delta j to be capital integral of delta x law to the J x over x squared DX, then a to j can be written as a difference of these two. Delta J's and writing it in this fashion allows us to bring in the gear polynomials will occur polynomial if you're not familiar. Is summation and choose J minus x to the power J over J factor of the beautiful polynomial, and then they're generalized linear polynomial as well, we'll come to that in a second, and it's not possible to rewrite as to then using the gear polynomials. So, the generalize the gear polynomials are going to be introduced soon. So as to a band can then be rewritten. You know the a to J's have now been written in terms of delta J's, and therefore you get these two terms minus and gamma zero T one of them to two and corresponding to these new coefficients, and T one of them can be written as an integral of the gear polynomials and T two of n can also be written as an integral involving the gear polynomials here. They're not quite linear polynomials are these generalized linear polynomials where you should do a shift on the binomial coefficient. So anyway, the bottom line is S two of n can be written as integrals of the gear polynomials. And the point is that the, the formula when we write it in that fashion. We get that S two of n is minus and gamma zero times minus and one to infinity delta of x or x squared, plus these two terms involving these generalized the gear polynomial one to infinity delta of x or x squared. Ln minus one, Ln minus two value log x. This can be further simplified to give that S two of n is actually minus and gamma zero. Plus the integral one to infinity delta of x or x squared Ln minus one. Two log x minus and one to infinity delta of x over x squared DX. Now, interestingly enough, so S two of n has now broken up into three parts of this is obviously innocuous, given the Bombardia legarious criterion, and this also equally can be shown to be completely innocuous, given the error term on the prime number theorem so last term in the ability to actually own by simple back application of the unconditional error term the prime, prime number theorem. To prove the remuner process therefore we need to focus on the interval in the second term. So, so that's what that's where I would like to stop I just want to kind of make a give a quick prediction word is but the future directions let's put it that way, somatic future directions, the most fascinating aspect of these criterion and the companion arithmetic formulas is that the behavior of zeta s at s equals one determines the location of all the non trivial zero and that very interesting. Lease criterion has been studied for number fields. It's been studied for the Selber class by a galaxy of authors. I just want to draw your attention to one particular paper by Francis Brown, who did the lead criterion in the number field context for dedicated data functions and notice that the positivity of lambda to which was child's play is that the remuner data function simply because it was the interpreters of variants of a probability density function and therefore not negative. In the dedicated data function case. It's not so simple we do not have a probabilistic interpretation of the dedicated data function. So, you know, I, you know, the end and the company, we have small cases, I believe they do some. Maybe the calcium ring or something like that something like that, some small number fields in which they look at that. And they're, they, they say similar things can be done. In general, to do the number field situation. This. I find this fascinating lambda to the positivity of lambda to for the dedicated data function, of course hinges on the fact that can you repeat Polia's calculation of interpreting the dedicated function or appropriate version of it. It's modified by gamma functions and so forth, as the melon transform of a non negative probability distribution. So that still seems to be quite open. So if one had a probabilistic view of the dedicated data function that could lead to the solution will see the zero problem. Then we come to another theme so I'm giving you some branches of, you know, research that I think are fascinating. The White House and others have studied what's called a modified Lee criterion. So if I was interested in a quasi ring hypothesis nose heroes to the right of Sigma, can I establish a similar criterion and the answer is yes you can. You can do that so the whole program of Lee as well as the program of Bombay or legerious can be transplanted to study the quasi. So as far as as far as I'm aware, the study of the probabilistic interpretation of the quasi GRH Lee criterion in terms of probability theory has not been fully unraveled. The nature of the still just constant is again of interest in transcendental number three as well as most of you know the Euler constant is known is not known to be transcendental is not even going to be rational. However, I find this very fascinating that the these Euler constants show up. And there's a nice paper by the horror and Kumar movie on what's called the Euler chronicle constants and these again are essentially steltches type constants that arise in the context of the dedicated data function that seem to play a very role. So it seems to me that there's a big program of studying these constants in the large in kind of very abstract sort of way. So it would be of interest to review many of these results. In fact, review the studies of still just constants for number fields, silver class automorphical functions and so on so forth, in terms of a probabilistic angle. So with that, I, I thank you for your attention. Thank you.