Partitions into Sums of Two Squares

TL;DR

This paper develops an asymptotic formula for counting partitions of integers into sums of two squares using advanced analytic techniques, addressing complexities from fractional singularities in the associated Dirichlet series.

Contribution

It introduces a novel application of the Circle and Saddle Point Methods to partitions into sums of two squares, handling fractional singularities in the Dirichlet series.

Findings

Derived an asymptotic formula for the number of partitions into sums of two squares

Extended analytic methods to handle fractional singularities in Dirichlet series

Provided insights into the distribution of such partitions for large n

Abstract

We use a variation of the Circle Method, along with the Saddle Point Method, to obtain an asymptotic formula for the number of partitions of a number n into integers which are sums of two squares. Unlike previous work on partitions into restricted parts, we need to handle a Dirichlet series with a fractional singularity