Some problems in analytic number theory for polynomials over a finite field

TL;DR

This paper discusses various problems in analytic number theory for polynomials over finite fields, presenting new approaches that can verify and inspire conjectures related to primes, Mobius function, and divisor problems.

Contribution

It introduces alternative methods for analytic number theory in function fields, enabling verification of existing conjectures and proposing new ones.

Findings

Counting primes in short intervals and progressions

Results related to Chowla's conjecture on Mobius autocorrelation

Insights into the additive divisor problem

Abstract

The lecture, given at the ICM 2014 in Seoul, explores several problems of analytic number theory in the context of function fields over a finite field, where they can be approached by methods different than those of traditional analytic number theory. The resulting theorems can be used to check existing conjectures over the integers, and to generate new ones. Among the problems discussed are: Counting primes in short intervals and in arithmetic progressions; Chowla's conjecture on the autocorrelation of the Mobius function; and the additive divisor problem.