# Intervals between numbers that are sums of two squares

## TL;DR

This paper improves estimates for the gaps between numbers representable as sums of two squares, using Bessel functions to bound moments of these gaps up to the second order.

## Contribution

It introduces new bounds for the moments of gaps between sums of two squares, extending previous results by employing sum of Bessel functions.

## Key findings

- Established upper bounds for the weighted mean values of gap moments
- Provided estimates for the $oldsymbol{ ext{γ}}$-th moments of gaps for all $ ext{γ} extless= 2$
- Enhanced understanding of the distribution of sums of two squares

## Abstract

In this paper, we improve the moment estimates for the gaps between numbers that can be represented as a sum of two squares of integers. We consider certain sum of Bessel functions and prove the upper bound for its weighted mean value. This bound provides estimates for the $\gamma$-th moments of gaps for all $\gamma\leq 2$.