Spring 2024 Schedule

• Tuesday, January 16, 2024 , 4 pm PST (note the unusual day and time)

Chiara Bellotti, University of New South Wales, Canberra

      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
 

• January 22, 2024

     Winston Heap, NTNU (Norwegian University of Science and Technology) 

     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   

• January 29, 2024

Emily Quesada-Herrera, TU Graz (Graz University of Technology)

     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 

• February 5, 2024

Eugenia Rosu, Mathematical Institute, Universiteit Leiden

     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.

• February 12, 2024

     Vivian Kuperberg, ETH Zurich

     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 

• February 26, 2024

     Bittu, IIIT (Indraprastha Institute of Information Technology), Delhi

     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

• March 4, 2024

Julia Stadlmann, University of Oxford

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 

• March 11, 2024 Cancelled

Sneha Chaubey, IIIT (Indraprastha Institute of Information Technology), Delhi

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. 

• March 18, 2024

     Quanli Shen, Shandong University, Weihai

     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 

• March 25, 2024

    Jérémy Dousselin, Institut Élie Cartan de Lorraine, Nancy
    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

• Wednesday, April 10, 2024 (note the unusual day)

Greg Knapp, University of Calgary

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

Seminar Organizers

If you are interested in giving a talk in the seminar that is related to the themes of this Collaborative Research Group, feel free to contact any of the organizers.