Beyond the LSD method for the partial sums of multiplicative functions

Andrew Granville,Dimitris Koukoulopoulos

doi:10.1007/s11139-018-0119-3

Andrew Granville, Dimitris Koukoulopoulos

Open Access

https://doi.org/10.1007/s11139-018-0119-3

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Abstract

The Landau–Selberg–Delange 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.

Highlights

Let f be a multiplicative function whose prime values are α on average, where α denotes a fixed complex number
Tenenbaum’s book [8] contains a detailed description of the LSD method along with a general theorem that evaluates the partial sums of multiplicative functions f satisfying a certain set of axioms
Our goal in this paper is to prove an appropriate version of the above asymptotic formula under the weaker condition f ( p) log p = αx + O

Summary

Introduction

Let f be a multiplicative function whose prime values are α on average, where α denotes a fixed complex number. In its modern form, it permits us to establish a precise asymptotic expansion for the partial sums of τα and for more general multiplicative functions. Tenenbaum’s book [8] contains a detailed description of the LSD method along with a general theorem that evaluates the partial sums of multiplicative functions f satisfying a certain set of axioms. In the series [9,10], obtained estimates for the partial sums of f under the weaker hypothesis p≤x ( f ( p) − α) = o(x/ log x) as x → ∞, together with various technical conditions ensuring that the values of f ( p)/α are restricted in an appropriate part of the complex plane (these conditions are automatically met if f ≥ 0, for example).

Initial preparations

Using Perron’s formula

Proof of Theorem 1

The error term in Theorem 1 is necessary