On the Distribution of Products of Primes and Powers

TL;DR

This paper investigates the distribution of numbers that are products of a prime and a k-th power, providing asymptotic formulas, error term improvements under the Riemann hypothesis, and results in short intervals.

Contribution

It generalizes Cohen's result by deriving new asymptotic formulas and error estimates for these numbers, including in short intervals, under the Riemann hypothesis.

Findings

Asymptotic formula for counting such numbers

Error term can be improved assuming Riemann hypothesis

Asymptotic results in short intervals

Abstract

We prove several results regarding the distribution of numbers that are the product of a prime and a $k$-th power. First, we prove an asymptotic formula for the counting function of such numbers; this generalises a result of E. Cohen. We then show that the error term in this formula can be sharpened on the assumption of the Riemann hypothesis. Finally, we prove an asymptotic formula for these counting functions in short intervals.