Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields
Number Theory
2014-03-24 v3
Abstract
For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n\le x such that q does not divide phi(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the prime k-tuples conjecture and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be both true.
Keywords
Cite
@article{arxiv.1108.3805,
title = {Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields},
author = {Kevin Ford and Florian Luca and Pieter Moree},
journal= {arXiv preprint arXiv:1108.3805},
year = {2014}
}
Comments
v3, final version. 31 pages. A few small typos and misprints corrected. To appear in Math. Comp