Asymptotic Estimates for Some Number Theoretic Power Series

TL;DR

This paper derives sharper asymptotic bounds for generating functions of key arithmetic functions using advanced zero-free region estimates of the Riemann zeta-function, improving upon classical Abelian theorem bounds.

Contribution

It introduces new asymptotic bounds for power series of arithmetic functions based on the Korobov-Vinogradov zero-free region, surpassing previous Abelian theorem estimates.

Findings

Sharper asymptotic bounds for Moebius, Liouville, and von Mangoldt functions' generating series.

Utilization of the Korobov-Vinogradov zero-free region for improved estimates.

Comparison showing these bounds are more precise than classical Abelian bounds.

Abstract

We derive asymptotic bounds for the ordinary generating functions of several classical arithmetic functions, including the Moebius, Liouville, and von Mangoldt functions. The estimates result from the Korobov-Vinogradov zero-free region for the Riemann zeta-function, and are sharper than those obtained by Abelian theorems from bounds for the summatory functions.