Short intervals asymptotic formulae for binary problems with primes and powers, II: density $1$

TL;DR

This paper establishes asymptotic formulas for representing integers as sums involving prime squares and squares in short intervals, both assuming the Riemann Hypothesis and unconditionally.

Contribution

It provides new short interval asymptotic results for binary problems with primes and powers, extending previous work to the density 1 case.

Findings

Asymptotic formulas hold for sums of prime squares and squares in short intervals

Results are valid both under the Riemann Hypothesis and unconditionally

Advances understanding of prime power representations in number theory

Abstract

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.