A Ces\`aro Average of generalised Hardy-Littlewood numbers

TL;DR

This paper develops a Cesàro weighted explicit formula for generalized Hardy-Littlewood numbers, extending previous work on additive prime problems and providing new insights into representations involving prime powers and squares.

Contribution

It introduces a Cesàro averaging approach to derive explicit formulas for generalized Hardy-Littlewood numbers, advancing the understanding of their distribution.

Findings

Derived a Cesàro weighted explicit formula for these numbers

Extended previous results on additive problems with primes

Improved understanding of representations involving prime powers and squares

Abstract

We continue our recent work on additive problems with prime summands: we already studied the \emph{average} number of representations of an integer as a sum of two primes, and also considered individual integers. Furthermore, we dealt with representations of integers as sums of powers of prime numbers. In this paper, we study a Ces\`aro weighted partial \emph{explicit} formula for generalised Hardy-Littlewood numbers (integers that can be written as a sum of a prime power and a square) thus extending and improving our earlier results.