Values of the Euler phi function not divisible by a prescribed odd prime

2006/11/16 by Pieter Moree, Moree, Pieter
Mathematics · #11N37 #11Y60 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11N37 #msc:11Y60

paper · pdf · doi:10.48550/arxiv.math/0611509

20 pages, 2 tables (each of them page filling)

arxiv created 2006/11/16 · openalex publication_date 2006/11/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let phi denote Euler's phi function. For a fixed odd prime we give an asymptotic series expansion in the sense of Poincare for the number E_q(x) of n<=x such that q does not divide phi(n). Thereby we improve on a recent theorem of B.K. Spearman and K.S. Williams [Ark. Mat. 44 (2006), 166-181]. Furthermore we resolve, under the Generalized Riemann Hypothesis, which of two approximations to E_q(x) is asymptotically superior using recent results of Y. Ihara on the Euler-Kronecker constant of a number field.

Citations

• Approximate formulas for some functions of prime numbers

Related

• Speculations on some characteristic properties of numbers
• On the irregular primes with respect to Euler polynomials
• Asymptotic evaluation of Euler-phi sums of various residue classes
• The distribution of gcd(n,ϕ(n))
• Simultaneous inequalities among values of the Euler phi-function