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

Abstract

Let ϕ \phi denote Euler’s phi function. For a fixed odd prime q q we investigate the first and second order terms of the asymptotic series expansion for the number of n ⩽ x n\leqslant x such that q ∤ ϕ ( n ) q\nmid \phi (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 k -tuples and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be both true.