Winter 2025 Talks
Title: Non-vanishing for cubic Hecke L-functions
Abstract: I will discuss recent work with Chantal David, Alexander Dunn, and Joshua Stucky, in which we prove that a positive proportion of Hecke L-functions associated to the cubic residue symbol modulo square-free Eisenstein integers do not vanish at the central point. Our principal new contribution is the asymptotic evaluation of the mollified second moment. No such asymptotic formula was previously known for a cubic family (even over function fields).
Our new approach makes crucial use of Patterson's evaluation of the Fourier coefficients of the cubic metaplectic theta function, Heath-Brown's cubic large sieve, and a Lindelöf-on-average upper bound for the second moment of cubic Dirichlet series that we establish. The significance of our result is that the family considered does not satisfy a perfectly orthogonal large sieve bound. This is quite unlike other families of Dirichlet L-functions for which unconditional results are known (namely the family of quadratic characters and the family of all Dirichlet characters modulo q). Consequently, our proof has fundamentally different features from the corresponding works of Soundararajan and of Iwaniec and Sarnak.
Recording
Title: Effective equidistribution of Galois orbits for mildly regular test functions
Abstract: We provide a detailed study on effective versions of the celebrated Bilu's equidistribution theorem for Galois orbits of sequences of points of small height in the N-dimensional algebraic torus, identifying the qualitative dependence of the convergence in terms of the regularity of the test functions considered. We develop a general Fourier analysis framework that extends previous results obtained by Petsche (2005), and by D'Andrea, Narváez-Clauss and Sombra (2017). This is a joint work with Mithun Das (ICTP).
Title: Convolution sums from Trace Formulae
Abstract: Previously we found certain convolution sums of divisor functions arising from physics yield Fourier coefficients of modular forms. In this talk we will discuss the limitations of the current proof of these formulas. We will also explore the connection with the Petersson and Kuznetsov Trace Formulae and the possibility of extending these formulas to other cases. The work mentioned in this talk is in collaboration with Ksenia Fedosova, Stephen D. Miller, Danylo Radchenko, and Don Zagier.
Recording
February 4, 2025 Dmitry Frolenkov, HSE University (National Research University, Higher School of Economics)
(This talk will take place at 10 am Pacific time. Note the unusual time.)
Title: Moments of symmetric square L-functions
Abstract: I am going to discuss various results on moments of symmetric square L-functions and some of their applications. I will mainly focus on a recent result of R. Khan and M. Young and our improvement of it. Khan and Young proved a mean Lindelöf estimate for the second moment of Maass form symmetric-square L-functions L(sym² uⱼ, 1/2 + it) on the short interval of length G ≫ |tⱼ|^(1 + ε)/t^(2/3), where tⱼ is a spectral parameter of the corresponding Maass form. Their estimate yields a subconvexity estimate for L(sym² uⱼ, 1/2 + it) as long as |tⱼ|^(6/7 + δ) ≪ t < (2 - δ)|tⱼ|. We obtain a mean Lindelöf estimate for the same moment in shorter intervals, namely for G ≫ |tⱼ|^(1 + ε)/t. As a corollary, we prove a subconvexity estimate for L(sym² uⱼ, 1/2 + it) on the interval |tⱼ|^(2/3 + δ) ≪ t ≪ |tⱼ|^(6/7 - δ). This is joint work with Olga Balkanova.
Recording
Title: Euler products inside the critical strip
Abstract: Even though Euler products of L-functions are generally valid only to the right of the critical strip, there is a strong sense in which they should persist even inside the critical strip. Indeed, the behaviour of Euler products inside the critical strip is very closely related to several major problems in number theory including the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture. In this talk, we will give an introduction to this topic and then discuss recent work on establishing asymptotics for partial Euler products of L-functions in the critical strip. We will end by giving applications of these results to questions related to Chebyshev's bias.
Recording
Title: Higher-order Titchmarsh problem via exceptional zeros
Abstract: The higher-order Titchmarsh problem concerns the correlation between higher divisor functions and primes. In this talk, I will explain how to derive an asymptotic formula for this correlation in appropriate ranges, assuming the existence of a "strong" Landau-Siegel zero. If time permits, I will also briefly discuss my ongoing work on further illusory consequences.
Title: Almost sure bounds for sums of random multiplicative functions
Abstract: I will start with a survey on sums of random multiplicative functions, focusing on distributional questions and almost sure upper bounds and Ω-results. In this context, I will describe previous work with Jake Chinis on a central limit theorem for correlations of Rademacher multiplicative functions, as well as ongoing work on establishing almost sure sharp bounds for them.
Recording
Title: Explicit Deuring-Heilbronn phenomenon for Dirichlet L-functions
Abstract: Deuring-Heilbronnn phenomenon, quantitatively established by Linnik in 1944, describes how the existence of a Landau-Siegel zero, which is real and near s=1, affects the location of the rest of the zeros of the Dirichlet L-functions to the same modulus. In this talk, we discuss an explicit version of this phenomenon based on our work initiated in the summer school ''Inclusive Paths in Explicit Number Theory'' with Asif Zaman, Shivani Goel, and Henry Twiss.
(This talk will take place at 10 am Pacific time. Note the unusual time.)
Title: A Survey on the Evaluation of Dirichlet L-Functions and Their Logarithmic Derivatives on the Line Re(s)=1
Abstract: The values of Dirichlet L-functions at s=1 have long attracted considerable attention due to their deep algebraic and geometric significance. In contrast, the logarithmic derivatives of Dirichlet L-functions at s=1, which play a key role in the study of prime distribution, remain less thoroughly understood despite their importance, a topic of interest since Dirichlet's groundbreaking work in 1837.
In this talk, we survey known results on the evaluation of Dirichlet L-functions and their logarithmic derivatives at s=1+it0, for a fixed real number t0.
(This talk will take place at 10 am Pacific time. Note the unusual time.)
Title: Zeros of L-functions in low-lying intervals and de Branges spaces
Abstract: We consider a variant of a problem first introduced by Hughes and Rudnick (2003) and generalized by Bernard (2015) concerning conditional bounds for small first zeros in a family of L-functions. Here we seek to estimate the size of the smallest intervals centered at a low-lying height for which we can guarantee the existence of a zero in a family of L-functions. This leads us to consider an extremal problem in analysis which we address by applying the framework of de Branges spaces, introduced in this context by Carneiro, Chirre, and Milinovich (2022).
Recording
Fall 2024 Talks
Andrés Chirre, Pontificia Universidad Católica del Perú
Title: Remarks on a formula of Ramanujan
Abstract: In this talk, we will discuss a well-known formula of Ramanujan and its relationship with the partial sums of the Möbius function. Under some conjectures, we analyze a finer structure of the involved terms. It is a joint work with Steven M. Gonek (University of Rochester).
Recording
Andrew Yang, University of New South Wales Canberra, ADFA
Title: Explicit zero-free regions or the Riemann zeta-function for large t
Abstract: A zero-free region of the Riemann zeta-function is a subset of the complex plane where the zeta-function is known to not vanish. In this talk we will discuss various computational and analytic techniques used to enlarge the zero-free region for the Riemann zeta-function, when the imaginary part of a complex zero is large. We will also explore the limitations of currently known approaches. This talk will reference a number of works from the literature, including a joint work with M. Mossinghoff and T. Trudgian.
Recording
Christine K. Chang, City University of New York
Title: Hybrid Statistics of the Maxima of a Random Model of the Zeta Function over Short Intervals
Abstract: We will present a matching upper and lower bound for the right tail probability of the maximum of a random model of the Riemann zeta function over short intervals. In particular, we show that the right tail interpolates between that of log-correlated and IID random variables as the interval varies in length. We will also discuss a new normalization for the moments over short intervals. This result follows the recent work of Arguin Dubach-Hartung and is inspired by a conjecture by Fyodorov-Hiary-Keating on the local maximum over short intervals.
Recording
Saloni Sinha, University of Missouri
Title: A Study of Twisted Sums of Arithmetic Functions
Abstract: We study sums of the form ∑n≤x f(n) n⁻ⁱʸ, where f is an arithmetic function, and we establish an equivalence between the Riemann Hypothesis and estimates on these sums. In this talk, we will explore examples of such sums with specific arithmetic functions, as well as discuss potential implications and future research directions.
Recording
Martin Čech, Univerzita Karlova (Charles University of Prague)
(This talk will take place at 11 am Pacific time on a Wednesday. Note the unusual day and time.)
Title: Moments of real Dirichlet L-functions and multiple Dirichlet series
Abstract: There are two ways to compute moments in families of L-functions: one uses the approximation by Dirichlet
polynomials, and the other is based on multiple Dirichlet series. We will introduce the two methods to study the family of
real Dirichlet L-functions, compare them and show that they lead to the same results. We will then focus on obtaining the
meromorphic continuation of the associated multiple Dirichlet series.
Recording
Henry Twiss, Brown University
Title: Subconvexity for GL(2) L-functions and Shifted MDS
Abstract: Subconvexity problems have maintained extreme interest in analytic number theory for decades. Critical barriers such as the convexity, Burgess, and Weyl bounds hold particular interest because one usually needs to drastically adjust the analytic techniques involved in order to break through them. It has recently come to light that shifted Dirichlet series can be used to obtain subconvexity results. While these Dirichlet series do not admit Euler products, they are amenable to study via spectral methods. In this talk, we construct a shifted multiple Dirichlet series (MDS) and leverage its analytic continuation via spectral decompositions to improve upon the Burgess bound in the conductor-aspect for the L-function of a holomorphic cusp form twisted by an arbitrary Dirichlet character. This improves upon the corresponding bound obtained by Blomer-Harcos in 2008. This work is joint with Jeff Hoffstein, Nikos Diamantis, and Min Lee.
Recording
Jackie Voros, University of Bristol
Title: On the average least negative Hecke eigenvalue
Abstract: The least quadratic non-residue has been a central problem in number theory for centuries. The average least quadratic non-residue was explored by Erdős in the 1960s, and many extensions of this problem such as to the average least character non-residue (Martin, Pollack) have been explored. In this talk, we look in to the average first sign change of Fourier coefficients of newforms (equivalently Hecke eigenvalues). We discuss the distribution of Hecke eigenvalues through the so-called 'horizontal' and 'vertical' Sato-Tate distributions, and we also discuss large sieve inequalities for cusp forms that are uniform in both the weight and the level.
Recording & Slides
Valeriya Kovaleva, Université de Montréal
Title: Correlations of the Riemann Zeta on the Critical Line
Abstract: In this talk we will discuss the behaviour of the Riemann zeta on the critical line, and in particular, its correlations in various ranges. We will prove a new result for correlations of squares, where shifts may be up to size T3/2-ε. We will also explain how this result relates to Motohashi’s formula for the fourth moment, as well as the moments of moments of the Riemann Zeta and its maximum in short intervals.
Hung Bui, University of Manchester
Title: Mean values of Hardy's Z-function and weak Gram's laws
Abstract: We establish the fourth moments of the real and imaginary parts of the Riemann zeta-function, as well as the fourth power mean value of Hardy's Z-function at the Gram points. We also study two weak versions of Gram's law. We show that those weak Gram's laws hold a positive proportion of time. This is joint work with Richard Hall.
Recording & Slides
Nicol Leong, University of New South Wales Canberra, ADFA
Title: Explicit bounds for the logarithmic derivative and the reciprocal of the Riemann zeta function
Abstract: Bounds on the logarithmic derivative and the reciprocal of the Riemann zeta function are studied as they have a wide range of applications, such as computing bounds for Mertens function. In this talk, we are mainly concerned with explicit bounds. Obtaining decent bounds are tricky, as they are only valid in a zero-free region, and the constants involved tend to blow up as one approaches the edge of the region, and a potential zero. We will discuss such bounds, their uses, and the computational and analytic techniques involved. Finally, we also show how to obtain a power savings in the case of the reciprocal of zeta.
Recording
Winter 2024 Talks
Chiara Bellotti, University of New South Wales Canberra, ADFA
Title: Explicit bounds for ζ and a new zero free region
Abstract: In this talk, we prove that |ζ(σ+it)|≤ 70.7 |t|4.438(1-σ)^{3/2} log2/3|t| for 1/2≤ σ ≤ 1 and |t| ≥ 3, combining new explicit bounds for the Vinogradov integral with exponential sum estimates. As a consequence, we improve the explicit zero-free region for ζ(s), showing that ζ(σ+it) has no zeros in the region σ ≥ 1-1/(53.989 (log|t|)2/3(log log|t|)1/3) for |t| ≥ 3.
Recording
Winston Heap, NTNU (Norwegian University of Science and Technology)
Title: Mean values of long Dirichlet polynomials
Abstract: We discuss the role of long Dirichlet polynomials in number theory. We first survey some applications of mean values of long Dirichlet polynomials over primes in the theory of the Riemann zeta function which includes central limit theorems and pair correlation of zeros. We then give some examples showing how, on assuming the Riemann Hypothesis, one can compute asymptotics for such mean values without using the Hardy-Littlewood conjectures for additive correlations of the von-Mangoldt functions.
Recording
Emily Quesada-Herrera, TU Graz (Graz University of Technology)
Title: Fourier optimization and the least quadratic non-residue
Abstract: We will explore how a Fourier optimization framework may be used to study two classical problems in number theory involving Dirichlet characters: The problem of estimating the least character non-residue; and the problem of estimating the least prime in an arithmetic progression. In particular, we show how this Fourier framework leads to subtle, but conceptually interesting, improvements on the best current asymptotic bounds under the Generalized Riemann Hypothesis, given by Lamzouri, Li, and Soundararajan. Based on joint work with Emanuel Carneiro, Micah Milinovich, and Antonio Ramos.
Recording
Eugenia Rosu, Mathematical Institute, Universiteit Leiden
Title: A higher degree Weierstrass function
Abstract: The Weierstrass ℘-function plays a great role in the classic theory of complex elliptic curves. A related function, the Weierstrass zeta-function, is used by Guerzhoy to construct preimages under the ξ -operator of newforms of weight 2, corresponding to elliptic curves. In this talk, I will discuss a generalization of the Weierstrass zeta-function and an application to harmonic Maass forms. More precisely, I will describe a construction of a preimage of the ξ-operator of a newform of weight k for k>2. This is based on joint work with C. Alfes-Neumann, J. Funke and M. Mertens.
Vivian Kuperberg, ETH Zurich
Title: Consecutive sums of two squares in arithmetic progressions
Abstract: In 2000, Shiu proved that there are infinitely many primes whose last digit is 1 such that the next prime also ends in a 1. However, it is an open problem to show that there are infinitely many primes ending in 1 such that the next prime ends in 3. In this talk, we'll instead consider the sequence of sums of two squares in increasing order. In particular, we'll show that there are infinitely many sums of two squares ending in 1 such that the next sum of two squares ends in 3. We'll show further that all patterns of length 3 occur infinitely often: for any modulus q, every sequence (a mod q, b mod q, c mod q) appears infinitely often among consecutive sums of two squares. We'll discuss some of the proof techniques, and explain why they fail for primes. Joint work with Noam Kimmel.
Recording
Bittu, IIIT Delhi (Indraprastha Institute of Information Technology)
Title: Spacing statistics of the Farey sequence
Abstract: The Farey sequence FQ of order Q is an ascending sequence of fractions a/b in the unit interval (0,1] such that (a,b)=1 and 0<a ≤ b ≤Q. The study of the Farey fractions is of major interest because of their role in problems related to the Diophantine approximation. Also, there is a connection between the distribution of Farey fractions and the Riemann hypothesis, which motivates their study. In this talk, we will discuss the distribution of Farey fractions with some divisibility constraints on denominators by studying their pair correlation measure. This is based on the joint work with Sneha Chaubey.
Recording
Julia Stadlmann, University of Oxford
Title: Primes in arithmetic progressions to smooth moduli
Abstract: The twin prime conjecture asserts that there are infinitely many primes p for which p+2 is also prime. This conjecture appears far out of reach of current mathematical techniques. However, in 2013 Zhang achieved a breakthrough, showing that there exists some positive integer h for which p and p+h are both prime infinitely often. Equidistribution estimates for primes in arithmetic progressions to smooth moduli were a key ingredient of his work. In this talk, I will sketch what role these estimates play in proofs of bounded gaps between primes. I will also show how a refinement of the q-van der Corput method can be used to improve on equidistribution estimates of the Polymath project for primes in APs to smooth moduli.
Recording
Sneha Chaubey, IIIT Delhi (Indraprastha Institute of Information Technology)
Title: Distribution of spacings of real-valued sequences
Abstract: The topic on the distribution of sequences saw its light with the seminal paper of Weyl. While the classical notion of equidistribution modulo one addresses the “global” behaviour of the fractional parts of a sequence, quantities such as k-point correlations and nearest neighbour gap distributions are useful in investigating the sequence on finer scales. In this talk, we discuss these fine-scale statistics for real-valued arithmetic sequences, and show that the limiting distribution of the nearest neighbour gaps of real-valued lacunary sequences is Poissonian. We also prove the Poissonian behavior of the 2-point correlation function for certain classes of real-valued vector sequences. This is achieved by extrapolating conditions on the number of solutions of Diophantine inequalities using twisted moments of the Riemann zeta function.
Quanli Shen, Shandong University, Weihai
Title: The fourth moment of quadratic Dirichlet L-functions
Abstract: I will discuss the fourth moment of quadratic Dirichlet L-functions where we prove an asymptotic formula with four main terms unconditionally. Previously, the asymptotic formula was established with the leading main term under generalized Riemann hypothesis. This work is based on Li's recent work on the second moment of quadratic twists of modular L-functions. It is joint work with Joshua Stucky.
Recording
Jérémy Dousselin, Institut Élie Cartan de Lorraine, Nancy
Title: Zeros of linear combinations of Dirichlet L-functions on the critical line
Abstract: Fix N≥ 1 and let L₁, L₂, ..., LN be Dirichlet L-functions with distinct, primitive and even Dirichlet characters. We assume that these functions satisfy the same functional equation. Let F(s)∶= c1L1(s)+c2L2(s)+…+cNLN(s) be a linear combination of these functions (cⱼ∈ℝ* are distinct). F is known to have two kinds of zeros: trivial ones, and non-trivial zeros which are confined in a vertical strip. We denote the number of non-trivial zeros 𝜌 with ℑ(𝜌)≤ T by N(T), and we let N₀(T) be the number of these zeros that are on the critical line. At the end of the 90's, Selberg proved that this linear combination had a positive proportion of zeros on the critical line, by showing that 𝜅F∶=𝑙iminfT (N₀(2T)−N₀(T))/(N(2T)−N(T))≥ c/N² for some c>0. Our goal is to provide an explicit value for c, and also to improve the lower bound above by showing that 𝜅F ≥ 2.16× 10⁻⁶/(N 𝑙og N), for any large enough N.
Recording
Greg Knapp, University of Calgary
Title: Bounds on the Number of Solutions to Thue Equations
Abstract: In 1909, Thue proved that when F(x,y) is an irreducible, homogeneous, polynomial with integer coefficients and degree at least 3, the inequality | F(x,y) | ≤ h has finitely many integer-pair solutions for any positive h. Because of this result, the inequality | F(x,y) | ≤ h is known as Thue’s Inequality. Much work has been done to find sharp bounds on the size and number of integer-pair solutions to Thue’s Inequality, with Mueller and Schmidt initiating the modern approach to this problem in the 1980s. In this talk, I will describe different techniques used by Akhtari and Bengoechea; Baker; Mueller and Schmidt; Saradha and Sharma; and Thomas to make progress on this general problem. After that, I will discuss some improvements that can be made to a counting technique used in association with “the gap principle” and how those improvements lead to better bounds on the number of solutions to Thue’s Inequality.
Recording
Ethan Lee, University of Bristol
Title: Conditional bounds for the error in the prime number theorem
Abstract: The prime number theorem describes the asymptotic distribution of the rational primes, and it is well known that we obtain the strongest bounds for the error in the prime number theorem when we assume the Riemann Hypothesis. Assuming the Riemann Hypothesis, the following two problems are natural to consider. First, are there refinements available for the error in the prime number theorem? Second, the prime number theorem indicates (asymptotically) how many primes we expect to see in a short interval, but how small is the error in this approximation? In this talk, I will (a) answer the first question in the affirmative and describe these refinements, and (b) introduce a smoothing argument to answer the second question.
Fall 2023 Talks
Cruz Castillo, University of Illinois Urbana-Champaign
Title: 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, ADFA
Title: 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
Neea Palojärvi, University of Helsinki
Title: 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
Title: 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
Title: 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
Title: The eighth moment of Γ_1(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 Γ_1(q) L-functions, which are associated with eigenforms for the congruence subgroups Γ_1(q). Similar to a large family of Dirichlet L-functions, the family of Γ_1(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
Title: 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
Title: 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
Title: 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
Winter 2023 Talks
Youness Lamzouri, Institut Élie Cartan de Lorraine
Title: Zeros of linear combinations of L-functions near the critical line
Abstract: 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
Title: Least quadratic non-residue and related problems
Abstract: 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, ADFA
Title: An explicit error term in the prime number theorem for large x
Abstract: 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
Title: The central value of Dirichlet L-functions over function fields and related topics
Abstract: A Dirichlet character over F_q(t) corresponds to a curve over F_q. 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 (Centre de Recherches Mathématiques)
Title: A logarithmic improvement in the Bombieri-Vinogradov theorem
Abstract: 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
Title: A uniform prime number theorem for arithmetic progressions
Abstract: 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
Title: A new explicit bound for the Riemann zeta function
Abstract: 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
Title: Euler's divergent series and primes in arithmetic progressions
Abstract: 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
Title: Products of primes in ray classes
Abstract: 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
Title: Exceptional Chebyshev's bias over finite fields
Abstract: 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
Title: The second moment of symmetric square L-functions over Gaussian integers
Abstract: 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 Talks
Akshaa Vatwani, Indian Institute of Technology Gandhinagar
Title: Joint extreme values of L-functions
Abstract: 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
Title: Joint value distribution of L-functions
Abstract: 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
Title: Races of irreducible monic polynomials in function fields
Abstract: 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
Title: Multiplicative functions in short intervals
Abstract: 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
Title: Extreme values of the Riemann zeta and Dirichlet L-functions at critical points
Abstract: 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
Title: Moments and Periods for GL(3)
Abstract: 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
Title: Uniform distribution and geometric incidence theory
Abstract: 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
Title: Quadratic Twists of Modular L-functions
Abstract: 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.
Recording Slides
Atul Dixit, Indian Institute of Technology Gandhinagar
Title: Voronoï summation formula for the generalized divisor function
Abstract: For a fixed z∈C and a fixed k∈N, let σ_z^(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 σ_z(n) as well as a divisor function d^(k)(n) first studied by Wigert. A Voronoï summation formula is obtained for σ_z^(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
Title: On non-Archimedean analogue of a question of Atkin and Serre
Abstract: 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
Title: The value distribution of the Hurwitz zeta function with an irrational shift
Abstract: 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
