CRG Weekly Seminar Series

Past Talks

In Spring 2023, the online seminars for the PIMS Collaborative Research Group L-functions in Analytic Number Theory were held on Wednesdays, 12-1 pm Pacific Time/ 1-2 pm Mountain Time and in Fall 2022, the seminars took place on Thursdays, 10–11 am Pacific Time/11 am–noon Mountain Time. 

Recorded talks are posted on our CRG page at mathtube. Individual links to the recordings of some previous seminars can also be found below.

Fall 2023 Schedule

    Cruz Castillo, University of Illinois Urbana-Champaign
     Sign changes of the error term in the Piltz divisor problem
Abstract: For an integer k≥3; Δk (x) :=∑n≤xdk(n)-Ress=1 (ζk(s)xs/s), where dk(n) is the k-fold divisor function, and ζ(s) is the Riemann zeta-function. In the 1950's, Tong showed for all large enough X; Δk(x) changes sign at least once in the interval [X, X + CkX1-1/k] for some positive constant Ck. For a large parameter X, we show that if the Lindelöf hypothesis is true, then there exist many disjoint subintervals of [X, 2X], each of length X1-1/k-ε such that Δk (x) does not change sign in any of these subintervals. If the Riemann hypothesis is true, then we can improve the length of the subintervals to << X1-1/k (logX)-k^2-2. These results may be viewed as higher-degree analogues of a theorem of Heath-Brown and Tsang, who studied the case k = 2. This is joint work with Siegfred Baluyot. 

     Recording

    Michaela Cully-Hugill, University of New South Wales Canberra
     An explicit estimate on the mean value of the error in the prime number theorem in intervals

Abstract: The prime number theorem (PNT) gives us the density of primes amongst the natural numbers. We can extend this idea to consider whether we have the asymptotic number of primes predicted by the PNT in a given interval. Currently, this has only been proven for sufficiently large intervals. We can also consider whether the PNT holds for sufficiently large intervals ‘on average’. This requires estimating the mean-value of the error in the PNT in intervals. A new explicit estimate for this will be given based on the work of Selberg in 1943, along with two applications: one for primes in intervals, and one for Goldbach numbers in intervals. 

     Recording 

   No seminar this week (Thanksgiving holiday)

   Neea Palojärvi, University of Helsinki
    Conditional estimates for logarithms and logarithmic derivatives in the Selberg class 

Abstract: The Selberg class consists of functions sharing similar properties to the Riemann zeta function. The Riemann zeta function is one example of the functions in this class. The estimates for logarithms of Selberg class functions and their logarithmic derivatives are connected to, for example, primes in arithmetic progressions.

In this talk, I will discuss about effective and explicit estimates for logarithms and logarithmic derivatives of the Selberg class functions when Re(s) ≥ 1/2+ 𝛿 where  𝛿 >0. All results are under the Generalized Riemann hypothesis and some of them are also under assumption of a polynomial Euler product representation or the strong λ-conjecture. The talk is based on a joint work with Aleksander Simonič (University of New South Wales Canberra).


     Recording


   Lucile Devin, Université du Littoral Côte d'Opale
   Biases in the distribution of Gaussian primes and other stories
Abstract: Generalizing the original Chebyshev bias can go in many directions: one can adapt the setting to virtually any equidistribution result encoded by a finite number of L-functions. In this talk, we will discuss what happens when one needs an infinite number of L-functions. This will be illustrated by the following question: given a prime that can be written as a sum of two squares p = a²+4b², how does the congruence class of a>0 distribute?


  Shivani Goel, Indraprastha Institute of Information Technology, Delhi
  On the Hardy Littlewood 3-tuple prime conjecture and convolutions of Ramanujan sums

Abstract: The Hardy and Littlewood k-tuple prime conjecture is one of the most enduring unsolved problems in mathematics. In 1999, Gadiyar and Padma presented a heuristic derivation of the 2-tuples conjecture by employing the orthogonality principle of Ramanujan sums.  Building upon their work, we explore triple convolution Ramanujan sums and use this approach to provide a heuristic derivation of the Hardy-Littlewood conjecture concerning prime 3-tuples. Furthermore, we estimate the triple convolution of the Jordan totient function using Ramanujan sums.

      

     Recording


   Vorrapan Chandee, Kansas State University

   The eighth moment of Γ1(q) L-functions

Abstract: In this talk, I will discuss my on-going joint work with Xiannan Li on an unconditional asymptotic formula for the eighth moment of Γ1(q) L-functions, which are associated with eigenforms for the congruence subgroups Γ1(q). Similar to a large family of Dirichlet L-functions, the family of  Γ1(q) L-functions has a size around q2 while the conductor is of size q. An asymptotic large sieve of the family is available by the work of Iwaniec and Xiaoqing Li, and they observed that this family of harmonics is not perfectly orthogonal. This introduces certain subtleties in our work. 


     Recording


   No seminar this week


    Andrew Pearce-Crump, University of York
    Characteristic polynomials, the Hybrid model, and the Ratios Conjecture 

Abstract: In the 1960s Shanks conjectured that the  ζ'(ρ), where ρ is a non-trivial zero of zeta, is both real and positive in the mean. Conjecturing and proving this result has a rich history, but efforts to generalise it to higher moments have so far failed. Building on the work of Keating and Snaith using characteristic polynomials from Random Matrix Theory, the Hybrid model of Gonek, Hughes and Keating, and the Ratios Conjecture of Conrey, Farmer, and Zirnbauer, we have been able to produce new conjectures for the full asymptotics of higher moments of the derivatives of zeta. This is joint work with Chris Hughes.


     Recording


   Siegfred Baluyot, American Institute of Mathematics

   Twisted moments of characteristic polynomials of random matrices

Abstract: In the late 90's, Keating and Snaith used random matrix theory to predict the exact leading terms of conjectural asymptotic formulas for all integral moments of the Riemann zeta-function. Prior to their work, no number-theoretic argument or heuristic has led to such exact predictions for all integral moments. In 2015, Conrey and Keating revisited the approach of using divisor sum heuristics to predict asymptotic formulas for moments of zeta. Essentially, their method estimates moments of zeta using lower twisted moments. In this talk, I will discuss a rigorous random matrix theory analogue of the Conrey-Keating heuristic. This is ongoing joint work with Brian Conrey.


     Recording


Sebastian Zuniga Alterman, University of Turku

     Möbius function, an identity factory with applications

Abstract: By using an identity relating a sum to an integral, we obtain a family of identities for the averages M(X)=∑n≤X µ(n)  and m(X)=∑n≤X µ(n)/n. Further, by choosing some specific families, we study two summatory functions related to the Möbius function, µ(n), namely ∑n≤X µ(n)/ns and ∑n≤X µ(n)/ns log(X/n), where s is a complex number and Re s >0. We also explore some applications and examples when s is real.  (joint work with O. Ramaré)

       Recording


Spring 2023 Schedule

  Youness Lamzouri, Institut Élie Cartan de Lorraine
   Zeros of linear combinations of L-functions near the critical line                                     
In this talk, I will present a recent joint work with Yoonbok Lee, where we investigate the number of zeros of linear combinations of L-functions in the vicinity of the critical line. More precisely, we let L1,…,LJ be distinct primitive L-functions belonging to a large class (which conjecturally contains all L-functions arising from automorphic representations on GL(n)), and b1,…,bJ be real numbers. Our main result is an asymptotic formula for the number of zeros of F(σ+it)=∑j≤JbjLj(σ+it) in the region σ≥1/2+1/G(T) and t∈[T,2T], uniformly in the range loglogT≤G(T)≤(logT)ν, where ν≍1/J. This establishes a general form of a conjecture of Hejhal in this range. The strategy of the proof relies on comparing the distribution of F(σ+it) to that of an associated probabilistic random model.

     Recording

  Enrique Treviño, Lake Forest College
   Least quadratic non-residue and related problems
In this talk we will talk about explicit estimates for character sums which have allowed us to find explicit estimates for the least quadratic non-residue and other related problems.

     Recording

  Daniel Johnston, University of New South Wales Canberra
  An explicit error term in the prime number theorem for large x
In 1896, the prime number theorem was established, showing that π(x) ∼  li(x). Perhaps the most widely used estimates in explicit analytic number theory are bounds on |π(x)-li(x)| or the related error term |θ(x)-x|. In this talk we discuss methods one can use to obtain good bounds on these error terms when x is large. Moreover, we will explore the many ways in which these bounds could be improved in the future.

     Recording

  Wanlin Li, Washington University in St. Louis
  The central value of Dirichlet L-functions over function fields and related topics

A Dirichlet character over Fq(t) corresponds to a curve over Fq. Using this connection to geometry, we construct families of characters whose L-functions vanish (resp. does not vanish) at the central point. The existence of infinitely many vanishing L-functions is in contrast with the situation over the rational numbers, where a conjecture of Chowla predicts there should be no such. Towards Chowla's conjecture, for each fixed q, we present an explicit upper bound on the number of such quadratic characters which decreases as q grows and it goes to 0 percent as q goes to infinity. In this talk, I will also discuss phenomena and interesting questions related to this problem. Some results in this talk are from projects joint with Ravi Donepudi, Jordan Ellenberg and Mark Shusterman.

  Alisa Sedunova, Saint Petersburg State University & CRM
   A logarithmic improvement in the Bombieri-Vinogradov theorem

We improve the best known to date result of Dress-Iwaniec-Tenenbaum, getting (log x)^2 instead of (log x)^(5/2). We use a weighted form of Vaughan's identity, allowing a smooth truncation inside the procedure, and an estimate due to Barban-Vehov and Graham related to Selberg's sieve. We give effective and non-effective versions of the result.

     Recording

  Asif Zaman, University of Toronto
  A uniform prime number theorem for arithmetic progressions

I will describe a version of the prime number theorem for arithmetic progressions that is uniform enough to deduce the Siegel-Walfisz theorem, Hoheisel's asymptotic for short intervals, a Brun-Titchmarsh bound and Linnik's bound for the least prime in an arithmetic progression. The proof combines Vinogradov-Korobov's zero-free region, a log-free zero density estimate and the Deuring-Heilbronn zero repulsion phenomenon. This is joint work with Jesse Thorner.

  Ghaith Hiary, Ohio State University
   A new explicit bound for the Riemann zeta function


I give a new explicit bound for the Riemann zeta function on the critical line. This is joint work with Dhir Patel and Andrew Yang. The context of this work highlights the importance of reliability and reproducibility of explicit bounds in analytic number theory. 


     Recording

  Anne-Maria Ernvall-Hytönen, University of Helsinki
  Euler's divergent series and primes in arithmetic progressions 

Euler's divergent series ∑_{n>0} n! z^n which converges only for z = 0 becomes an interesting object when evaluated with respect to a p-adic norm (which will be introduced in the talk). Very little is known about the values of the series. For example, it is an open question whether the value at one is irrational (or even non-zero). As individual values are difficult to reach, it makes sense to try to say something about collections of values over sufficiently large sets of primes. This leads to looking at primes in arithmetic progressions, which is in turn raises a need for an explicit bound for the number of primes in an arithmetic progression under the generalized Riemann hypothesis.

During the talk, I will speak about both sides of the story: why we needed good explicit bounds for the number of primes in arithmetic progressions while working with questions about irrationality, and how we then proved such a bound.

The talk is joint work with Tapani Matala-aho, Neea Palojärvi and Louna Seppälä. (Questions about irrationality with T. M. and L. S. and primes in arithmetic progressions with N. P.)

  Jyothsnaa Sivaraman, Chennai Mathematical Institute

  Products of primes in ray classes

In 1944, Linnik showed that the least prime in an arithmetic progression given by a mod q for (a,q)=1 is at most cq^L for some absolutely computable constants c and L. A lot of work has gone in computing explicit bounds for c and L. The best known bound is due to Xylouris (2011) who showed that c can be taken to be 1 and L to be 5 for q sufficiently large. In 2018, Ramaré and Walker gave a completely explicit result if one prime is replaced by a product of primes. They showed that each co-prime class modulo q contains a product of three primes each less than q^(16/3). This was improved by Ramaré, Srivastava and Serra to 650q^3 in 2020. In this talk we will introduce analogous results in the set up of narrow ray class fields of number fields. This is joint work with Deshouillers, Gun and Ramaré.

  Alexandre Bailleul, ENS Paris-Saclay

  Exceptional Chebyshev's bias over finite fields 

Chebyshev's bias is the surprising phenomenon that there is usually more primes of the form 4n+3 than of the form 4n+1 in initial intervals of the natural numbers. More generally, following work from Rubinstein and Sarnak, we know Chebyshev's bias favours primes that are not squares modulo a fixed integer q compared to primes which are squares modulo q. This phenomenon also appears over finite fields, where we look at irreducible polynomials modulo a fixed polynomial M. However, in the finite field case, there are a few known exceptions to this phenomenon, appearing as a result of multiplicative relations between zeroes of certain L-functions. In this work, we show,  improving on earlier work by Kowalski, that those exceptions are rare. This is joint work with L. Devin, D. Keliher and W. Li.

  No Seminar this week

  Olga Balkanova, Steklov Mathematical Institute
   The second moment of symmetric square L-functions over Gaussian integers


We prove an explicit formula for the first moment of Maass form symmetric square L-functions defined over Gaussian integers. As a consequence, we derive a new upper bound for the second moment. This is joint work with Dmitry Frolenkov.

Fall 2022 Schedule

 Akshaa Vatwani, Indian Institute of Technology Gandhinagar

 Joint extreme values of L-functions   
We consider L-functions L1,...,Lk from the Selberg class having polynomial Euler product and satisfying Selberg’s orthonormality condition. We show that on every vertical line s=σ+it in the complex plane with σ∈(1/2,1), these L-functions simultaneously take “large” values inside a small neighborhood. Our method extends to σ=1 unconditionally, and to σ=1/2 on the generalized Riemann hypothesis. We also obtain similar joint omega results for arguments of the given L-functions. This is joint work with Kamalakshya Mahatab and Łukasz Pańkowski. 

    References                                

 Junxian Li, Mathematisches Institut der Universität Bonn  
  Joint value distribution of L-functions

It is believed that distinct primitive L-functions are “statistically independent”. The independence can be interpreted in many different ways. We are interested in the joint value distributions and their applications in moments and extreme values for distinct L-functions. We discuss some large deviation estimates in Selberg and Bombieri-Hejhal’s central limit theorem for values of several L-functions. On the critical line, values of distinct primitive L-functions behave independently in a strong sense. However, away from the critical line, values of distinct Dirichlet L-functions begin to exhibit some correlations.
This is based on joint works with Shota Inoue.
   References     Recording

 Youssef Sedrati, Institut Élie Cartan de Lorraine, Nancy

 Races of irreducible monic polynomials in function fields 
Chebyshev noticed in 1853 that there is a predominance, for “most” real numbers x≥2, of the number of primes ≤x and congruent to 3 modulo 4 over primes ≤x and congruent to 1 modulo 4. Since then, several generalizations of this phenomenon have been studied, notably in the case of prime number races with three or more competitors by Y. Lamzouri. In this talk, I will present results related to the generalization of Y. Lamzouri’s work in the context of polynomial rings over finite fields. I will also discuss results concerning races of irreducible monic polynomials involving two competitors. In particular, I will give examples where the races in the function field setting behave differently than in the number field setting.

     References

  Pranendu Darbar, Norwegian Institute of Science and Technology

  Multiplicative functions in short intervals

In this talk, we are interested in a general class of multiplicative functions. For a function that belongs to this class, we will relate its short average" to its "long average". More precisely, we will compute the variance of such a function over short intervals by using Fourier analysis and by counting rational points on certain binary forms. The discussion is applicable to some interesting multiplicative functions such as μk(n), ϕ(n)/n, n/ϕ(n), μ^2(n)ϕ(n)/n, σα(n), (−1)^#{p:pk|n}(n), and many others and it provides various new results and improvements to the previous result in the literature. This is a joint work with Mithun Kumar Das.

     References     Recording

 Shashank Chorge, University of Rochester
  Extreme values of the Riemann zeta and Dirichlet L-functions at critical points

We compute extreme values of the Riemann zeta function at the critical points of the zeta function in the critical strip. i.e. the points where ζ'(s)=0 and Rs<1. We show that the values taken by the zeta function at these points are very similar to the extreme values taken without any restrictions. We will show geometric significance of such points. We also compute extreme values of Dirichlet L- functions at the critical points of the zeta function to the right of Rs=1. It shows statistical independence of L-functions and zeta function in a certain way as these values are very similar to the values taken by L-functions without any restriction.

     References    Recording

  Chung-Hang (Kevin) Kwan, University College London

  Moments and Periods for GL(3)
In the past century, the studies of moments of L-functions have been important in number theory and are well-motivated by a variety of arithmetic applications. This talk will begin with two problems in elementary number theory, followed by a survey of techniques in the past and the present. We will slowly move towards the perspectives of period integrals which will be used to illustrate the interesting structures behind moments. In particular, we shall focus on the “Motohashi phenomena”.

     Recording

  Ayla Gafni, University of Mississippi

  Uniform  distribution and geometric incidence theory 

The Szemeredi-Trotter Incidence Theorem, a central result in geometric combinatorics, bounds the number of incidences between n points and m lines in the Euclidean plane. Replacing lines with circles leads to the unit distance problem, which asks how many pairs of points in a planar set of n points can be at a unit distance. The unit distance problem breaks down in dimensions 4 and higher due to degenerate configurations that attain the trivial bound. However, nontrivial results are possible under certain structural assumptions about the point set. In this talk, we will give an overview of the history of these and other incidence results. Then we will introduce a quantitative notion of uniform distribution and use that property to obtain nontrivial bounds on unit distances and point-hyperplane incidences in higher-dimensional Euclidean space. This is based on joint work with Alex Iosevich and Emmett Wyman.

  Xiannan Li, Kansas State University

  Quadratic Twists of modular L-functions

The behavior of quadratic twists of modular L-functions at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves. Here we describe a proof of an asymptotic for the second moment of this family of L-functions, previously available conditionally on the Generalized Riemann Hypothesis by the work of Soundararajan and Young. Our proof depends on deriving an optimal large sieve type bound.

     Slides   Recording  

  Atul Dixit, Indian Institute of Technology Gandhinagar

  Voronoï summation formula for the generalized divisor function
For a fixed z∈C and a fixed k∈N, let σz^(k)(n) denote the sum of z-th powers of those divisors d of n whose k-th powers also divide n. This arithmetic function is a simultaneous generalization of the well-known divisor function σz(n) as well as a divisor function d^(k)(n) first studied by Wigert. A Voronoï summation formula is obtained for σz^(k)(n). An interesting thing to note here is that this arithmetic function does not fall under the purview of the setting of the Hecke functional function with multiple gamma factors studied by Chandrasekharan and Narasimhan. Some applications of the Voronoï summation formula will be given. This is joint work with Bibekananda Maji and Akshaa Vatwani.

  Sanoli Gun, The Institute of Mathematical Sciences

  On non-Archimedean analogue of a question of Atkin and Serre

Let τ be the Ramanujan tau function. It is a well known question of Atkin and Serre that for any ϵ>0, there exists a constant c(ϵ)>0 such that
|τ(p)|≥c(ϵ)p^{(k−3)/2−ϵ}. In this talk, we will address a non-Archimedean analogue of this question which improves the recent bound of Bennett, Gherga, Patel and Siksek. This is a report on a joint work with Yuri Bilu and Sunil Naik.

 Anurag Sahay, University of Rochester

 The value distribution of the Hurwitz zeta function with an irrational shift 

The Hurwitz zeta function ζ(s, α) is a shifted integer analogue of the Riemann zeta function which shares many of its properties, but is not an ”L-function” under any reasonable definition of the word. We will first review the basics of the value distribution of the Riemann zeta function in the critical strip (moments, Bohr–Jessen theory...) and then contrast it with the value distribution of the Hurwitz zeta function. Our focus will be on shift parameters α /∈ Q, i.e., algebraic irrational or transcendental. We will present a new result (joint with Winston Heap) on moments of these objects on the critical line.

    Recording