COUNTING NUMBERS IN MULTIPLICATIVE SETS: LANDAU VERSUS RAMANUJAN

Author:

Pieter MOREE

2010 Mathematics Subject Classification:

11N37, 11Y60.

Keywords:

arithmetic progression, Euler-Kronecker constant, Landau approximation, multiplicative function, multiplicative set, Ramanujan approximation.

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 \leq 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).

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More-2013

Vol. 8 (16), 2013