Beyond the LSD method for the partial sums of multiplicative functions

TL;DR

This paper develops new techniques to estimate the partial sums of multiplicative functions when the prime average is known with weak error terms, extending beyond the classical LSD method.

Contribution

It introduces methods to derive partial sum estimates under weaker prime average error conditions than traditionally assumed.

Findings

New bounds for partial sums with weak prime average error

Extension of LSD method to broader error regimes

Improved understanding of multiplicative function behavior

Abstract

The Landau-Selberg-Delange (LSD) method gives an asymptotic formula for the partial sums of a multiplicative function $f$ whose prime values are $\alpha$ on average. In the literature, the average is usually taken to be $\alpha$ with a very strong error term, leading to an asymptotic formula for the partial sums with a very strong error term. In practice, the average at the prime values may only be known with a fairly weak error term, and so we explore here how good an estimate this will imply for the partial sums of $f$, developing new techniques to do so.