L-Functions in Analytic Number Theory
Analytic number theory focuses on arithmetic questions through the lens of L-functions. These generating series encode arithmetic information and have connections with a host of other mathematical fields, such as algebraic number theory, harmonic analysis, Diophantine approximation, probability, representation theory, and computational number theory. The main focuses of this CRG include moments of L-functions and automorphic forms, explicit results in analytic number theory, and comparative prime number theory.
Scientific, Seminar
L-functions in Analytic Number Theory: Greg Knapp
In 1909, Thue proved that when $F(x,y)$ is an irreducible, homogeneous, polynomial with integer coefficients and degree at least 3, the inequality $\left\| F(x,y) \right\| \leq h$ has finitely many integer-pair solutions for any positive $h$. Because...
Scientific, Seminar
L-functions in Analytic Number Theory: Jérémy Dousselin
Fix $N\geq 1$ and let $L_1, L_2, \ldots, L_N$ be Dirichlet L-functions with distinct, primitive and even Dirichlet characters. We assume that these functions satisfy the same functional equation. Let $F(s)∶= c_1L_1(s)+c_2L_2(s)+\ldots+c_NL_N(s)$ be a...
Scientific, Seminar
L-functions in Analytic Number Theory: Quanli Shen
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...
Scientific, Seminar
CANCELLED: L-functions in Analytic Number Theory: Sneha Chaubey
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...
Scientific, Seminar
L-functions in Analytic Number Theory: Julia Stadlmann
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...
Scientific, Seminar
L-functions in Analytic Number Theory: Biitu
Scientific, Seminar
L-functions in Analytic Number Theory: Vivian Kuperberg
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...
Scientific, Seminar
L-functions in Analytic Number Theory: Emily Quesada-Herrera
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...
Scientific, Seminar
L-functions in Analytic Number Theory: Winston Heap
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...
Scientific, Seminar
L-functions in Analytic Number Theory: Chiara Bellotti
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...