Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields

TL;DR

This paper studies the distribution of integers n where a fixed odd prime q does not divide Euler's totient function phi(n), and explores the behavior of Euler-Kronecker constants for cyclotomic fields, revealing incompatible conjectures.

Contribution

It provides an asymptotic analysis of the count of such n and demonstrates the mutual exclusivity of two prominent conjectures regarding Euler-Kronecker constants.

Findings

Asymptotic formulas for the count of n with q not dividing phi(n)

Analysis of Euler-Kronecker constants for cyclotomic fields

Proof that the prime k-tuples conjecture and Ihara's conjecture cannot both hold

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.