Values of the euler φ-function not divisible by a given odd prime, and the distribution of euler-kronecker constants for cyclotomic fields

Kevin Ford, Florian Luca, Pieter Moree

Research output: Contribution to journalArticlepeer-review

Abstract

Let φ denote Euler's φ function. For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n ≤ x such that q φ(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the Hardy-Littlewood conjecture about counts of prime k-tuples and a conjecture of Ihara about the distribution of these Euler- Kronecker constants cannot be both true.

Original languageEnglish (US)
Pages (from-to)1447-1476
Number of pages30
JournalMathematics of Computation
Volume83
Issue number287
DOIs
StatePublished - May 2014

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

Fingerprint

Dive into the research topics of 'Values of the euler φ-function not divisible by a given odd prime, and the distribution of euler-kronecker constants for cyclotomic fields'. Together they form a unique fingerprint.

Cite this