Short intervals asymptotic formulae for binary problems with prime powers, II

TL;DR

This paper refines asymptotic formulas for counting representations of integers as sums of prime powers within short intervals, advancing understanding of additive problems involving primes and fixed exponents.

Contribution

It improves existing asymptotic results for the average number of representations of integers as sums of prime powers in short intervals.

Findings

Enhanced asymptotic formulas for prime power representations

Broader applicability to fixed exponents in additive problems

Refined estimates for sums involving primes and prime powers

Abstract

We improve some results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell_1}+p_{2}^{\ell_2}$ and $n=p^{\ell_1} + m^{\ell_2}$, where $\ell_1, \ell_2\ge 2$ are fixed integers, $p,p_1,p_2$ are prime numbers and $m$ is an integer.