Counting numbers in multiplicative sets: Landau versus Ramanujan

TL;DR

This paper surveys the literature on counting numbers in multiplicative sets, comparing Landau and Ramanujan's approximations, and relates their asymptotic precision to Euler-Kronecker constants.

Contribution

It provides a comprehensive review of existing results on multiplicative sets and discusses the relationship between approximation methods and Euler-Kronecker constants.

Findings

Landau and Ramanujan's approximations are compared and related.

The asymptotic precision of these approximations is connected to Euler-Kronecker constants.

The paper emphasizes the author's contributions to this area.

Abstract

A set S of integers is said to be multiplicative if for every pair m and n of coprime integers we have that mn is in S iff both m and n are in S. Both Landau and Ramanujan gave approximations to S(x), the number of n<=x that are in S, for specific choices of S. The asymptotical precision of their respective approaches are being compared and related to Euler-Kronecker constants, a generalization of Euler's constant gamma=0.57721566.... This paper claims little originality, its aim is to give a survey on the literature related to this theme with an emphasis on the contributions of the author (and his coauthors).