Counting numbers in multiplicative sets: Landau versus Ramanujan

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).

15 pages, 1 Table

References in corpus (3)

• Principal forms X^2 + nY^2 representing many integers
• Experiments with moduli of quadrilaterals
• Values of the Euler phi function not divisible by a prescribed odd prime

Cited by in corpus (7)

• Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms
• Errata and Addenda to Mathematical Constants
• Topics In Primitive Roots
• Generalized Fibonacci Primitive Roots
• Generalized Artin Primitive Root Conjecture
• Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields
• Note on the Theory of Correlation Functions