Moments of Representation Numbers

TL;DR

This paper explores the asymptotic behavior of representation numbers, focusing on moments of ways to express integers as sums of squares and primes, using modern analytic number theory techniques.

Contribution

It applies recent methods to derive new bounds and results on moments of representation numbers involving sums of squares and primes.

Findings

Derived upper bounds on higher moments of representations as sum of a prime square and a square.

Obtained new results on moments of representation numbers with sums of two squares.

Extended existing techniques to broader classes of representation problems.

Abstract

A representation number is a function which expresses the number of ways an integer can be written as a sum of elements of chosen sets. One of the oldest number-theoretic results on representation numbers is Fermat's theorem which says that an odd prime can be written as a sum of two squares in exactly $0$ or $2$ ways (if order of summands is important). In this dissertation, we discuss a selection of methods from modern analytic number theory and apply them to study asymptotics of certain representation numbers. In particular, we work through an argument in the recent paper "The multiplication table constant and sums of two squares" by Granville, Sabuncu and Sedunova to obtain upper bounds on higher moments of the number of ways of writing $n$ as the sum of a square and a square of a prime. We then use the method from the paper to obtain new results on moments of representation numbers where ``prime'' is replaced by ``sum of two squares''.