Topics in Equidistribution and Exponential Sums

Citation

Shubin, Andrei (2022) Topics in Equidistribution and Exponential Sums. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/153e-5r72. https://resolver.caltech.edu/CaltechTHESIS:07162021-010608172

Abstract

In this thesis, we consider a few problems connected to the exponential sums which is one of the most important topics in analytic number theory.

In the first part, we study the distribution of prime numbers in special subsets of integers and, in particular, the distribution of these primes in arithmetic progressions, small gaps between them, the behavior of the corresponding exponential sums over primes, and related questions. Big progress was made on these questions in recent years. The famous works of Zhang and Maynard gave the proof of existence of bounded gaps between consecutive primes. Applying the sieve of Selberg-Maynard-Tao and an analogue of the Bombieri-Vinogradov theorem, we obtain similar results for a large class of subsets of primes and improve some of the previous results. The proof of the analogue of the Bombieri-Vinogradov theorem is also connected to a breakthrough work of Bourgain, Demeter, and Guth on the proof of Vinogradov Mean Value Conjecture via l²-decoupling. Their result, in particular, has led to a significant improvement of the classical van der Corput estimates for a large class of exponential sums.

In the second part, we study the behavior of higher moments of Gauss sum twisted by a Mobius function. The moments of exponential sums are very important in number theory and harmonic analysis as they appear in many other problems. The sum with the Mobius function is of independent interest because of the famous Sarnak Conjecture which is on the edge of number theory, analysis, and dynamical systems. The bound we obtain for L^p-norm of the sum confirms that the Mobius function is uncorrelated with the quadratic phase αn² for most α ϵ [0; 1].

In the third part, we study the distribution of lattice points on the surface of 3-dimensional sphere, which is known as Linnik problem. It turns out that the variance for such points is closely related to the behavior of certain GL(2) L-functions estimated at the central point 1/2. To evaluate the moments of these L-functions, we apply similar techniques used to evaluate the moments of Riemann zeta function on the critical line in the breakthrough works of Soundararajan and Harper. Their results have led to the sharp upper bounds for all positive moments of zeta function conditionally on Riemann Hypothesis and similar bounds for a broad class of L-functions in families conditionally on the corresponding Grand Riemann Hypothesis. We apply similar methods to get sharp upper bound for the variance of lattice points on the sphere. The connection of Weyl sums on the sphere to the sums of special values of GL(2) L-functions is a big output of the Langlands program, which has also gotten a lot of attention in recent years.

Thesis Files