# A note on an average additive problem with prime numbers

## TL;DR

This paper explores the average number of ways large integers can be expressed as sums of prime powers, analyzing how assumptions like the Riemann Hypothesis affect the results.

## Contribution

It advances understanding of prime power representations by examining average counts over short intervals under different hypotheses.

## Key findings

- Average representations depend on the interval length and hypotheses.
- Results differ significantly with or without assuming the Riemann Hypothesis.
- Provides refined estimates for the number of prime power sums.

## Abstract

We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we assume the Riemann Hypothesis.