Sums of almost equal squares of primes

TL;DR

This paper investigates the representation of large integers as sums of almost equal prime squares, improving bounds and extending results for different numbers of terms using sieve methods and number theory techniques.

Contribution

It introduces new bounds for representing integers as sums of prime squares with almost equal terms, extending previous work and providing first estimates for certain cases.

Findings

For s=5, all large integers ≡ 5 mod 24 are representable with θ > 8/9.

Improved estimates for integers with s=3,4 satisfying local conditions.

Extended previous results and provided first estimates for specific cases.

Abstract

We study the representations of large integers $n$ as sums $p_1^2 + ... + p_s^2$, where $p_1,..., p_s$ are primes with $| p_i - (n/s)^{1/2} | \le n^{\theta/2}$, for some fixed $\theta < 1$. When $s = 5$ we use a sieve method to show that all sufficiently large integers $n \equiv 5 \pmod {24}$ can be represented in the above form for $\theta > 8/9$. This improves on earlier work by Liu, L\"{u} and Zhan, who established a similar result for $\theta > 9/10$. We also obtain estimates for the number of integers $n$ satisfying the necessary local conditions but lacking representations of the above form with $s = 3, 4$. When $s = 4$ our estimates improve and generalize recent results by L\"{u} and Zhai, and when $s = 3$ they appear to be first of their kind.