The asymptotic behavior of limit-periodic functions on primes and an application to k-free numbers

TL;DR

This paper employs the circle method to analyze the average behavior of limit-periodic functions on primes, providing insights into the distribution of k-free numbers under linear shifts.

Contribution

It introduces a novel approach to evaluate limit-periodic functions on primes and applies this to study the distribution of k-free numbers with linear shifts.

Findings

Average value of certain limit-periodic functions on primes can be evaluated.

Distribution patterns of k-free numbers under linear shifts are characterized.

Conditions for the convergence of singular series are established.

Abstract

We use the circle method to evaluate the behavior of limit-periodic functions on primes. For those limit-periodic functions that satisfy a kind of Barban-Davenport-Halberstam condition and whose singular series converge fast enough, we can evaluate their average value on primes. As an application, this result is used to show how tuples of different k-free numbers behave when linear shifts are applied.