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 denote Euler's phi 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.

The smallest invariant factor of the multiplicative group

“…where Reg(K) and h(K) denote the regulator and class number of K. The form we have chosen has the advantage of being easy to estimate, as the following proposition will show. A very similar constant appears in the counting function of those integers n for which φ(n) is not divisible by a given prime q (see [6,2], and in particular note that our calculation (4.2) has a close counterpart in [2]). This is to be expected, as such integers are essentially those free of prime factors p ≡ 1 (mod q), which is the case B = {2, .…”

Euler’s function on products of primes in a fixed arithmetic progression

“…is the Euler-Kronecker constant associated with the cyclotomic field Qpe 2πi{p q. Computations of the value of γ p are provided in [2]. Using these and (51) we determine the value of F p for odd primes up to p " 149.…”

“…For an analogue of the prime number theorem, we have (2) θpx; q, aq " x ϕpqq , as x Ñ 8 (see [17,Theorem 6.8]), where θpx; q, aq :" ÿ pďx p"a pmod qq log p.…”

The distribution of Euler–Kronecker constants of quadratic fields

“…Murty [14] proved an upper bound for the first moment of γ K(q) , which was refined to an asymptotic formula by Fouvry [5], who showed that the average order of γ K(q) is log Q. In the case where q is prime, Ford, Luca and Moree [4] studied γ K(q) and showed that it appears in the asymptotic expansion of the number of integers n ≤ x for which ϕ(n) is not divisible by q, where ϕ is the Euler ϕ-function.…”