Student Analytic Number Theory
Organizers: Alex de Faveri, Jared Duker Lichtman, and Julia Stadlmann.
Please contact organizer for Zoom links.
Past Events
We will discuss a couple of recent papers of Walsh.
Note alternate location.
The dispersion method has found an impressive range of applications in analytic number theory, from bounded gaps between primes to the greatest prime factors of quadratic polynomials. The method requires bounding certain sums of Kloosterman sums, using deep inputs from algebraic geometry and the…
I will discuss joint work in progress with N. Arala, J. R. Getz, J. Hou, C.-H. Hsu, and H. Li, concerning a new, nonabelian circle method and its applications to counting problems of a classical flavor.
We report on recent and on-going work joint with Joerg Bruedern concerning problems involving the representation of integer sequences by sums of powers. Our new tool is an upper bound for moments of smooth Weyl sums restricted to wide major arcs. This permits progress to be made on Waring's…
Given a set of integers, we wish to know how many primes there are in the set. Modern tools allow us to obtain an asymptotic for the number of primes, or at least a lower bound of the expected order, assuming certain strength Type-I information (the distribution of the sequence in…
Choose your favourite, compact manifold M. How many rational points, with denominator of bounded size, are near M? We report on joint work with Damaris Schindler and Rajula Srivastava addressing this question. Our new method reveals an intriguing interplay between number theory, harmonic…
Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemerédi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.In this talk, we shall discuss various…
Quantum unique ergodicity (QUE) describes the equidistribution of the L2-mass of eigenfunctions of the Laplacian as their eigenvalues approach infinity. My focus lies on a specific variant known as holomorphic QUE, which concerns the distribution of the L2-mass of normalized…
Let r_0(n) be the number of representations of n as a sum of two squares, and r_1(n) count the number of representations of n as a sum of an integer square and a prime square. The asymptotic formulas for the moments of r_0(n), with k greater than 1 summed over n up to x are well-known via…
The spectral theory of automorphic forms finds remarkable applications in analytic number theory. Notably, it is utilised in results concerning the distribution of primes in large arithmetic progressions and in questions on variants of the fourth moment of the zeta function. Traditionally, these…