Values of the Euler phi function not divisible by a given odd prime
type: Item
instance of: scholarly article
title: Values of the Euler phi function not divisible by a given odd prime
zbMATH Open document ID: 1216.11089
author: Blair K. Spearman
author: Kenneth S. Williams
published in: Arkiv för Matematik
publication date: 2009-03-09T00:00:00Z
review text: Let \(\varphi(n)\) be the Euler function of the positive integer \(n\). This is always even if \(n\geq 3\). Given an odd prime \(q\), the authors determine an asymptotic formula for the number of positive integers \(n\leq x\) such that \(q\) does not divide \(\varphi(n)\). Denoting this number by \(E_q(x)\), the theorem on page 170 shows that \[ E_q(x)=e_0(q){{x}\over {(\log x)^{1/(q-1)}}}+O\Biggl({{x}\over {(\log x)^{q/(q-1)+\varepsilon}}}\Biggr) \] where \(\varepsilon\in (0,1)\) and the constant implied by the \(O\)-symbol depends on \(q\) and \(\varepsilon\). The authors gives an explicit formula for the positive constant \(e_0(q)\) involving invariants of the cyclotomic number field \({\mathbb K}(q)={\mathbb Q}(e^{2\pi i/q})\). The proof involves a Mertens type theorem for the product \[ \prod_{p\leq x,~p\equiv 1\pmod{q}}\Biggl(1-{1\over p}\Biggr). \] \{Reviewer's remark: The latter product can be derived in an easier way by invoking the Mertens' theorem for the algebraic number field \({\mathbb K}(q)\). Furthermore, a standard application of the Selberg-Delange method gives an asymptotic expansion \[ E_q(x)= {e_0(q)x\over (\log x)^{1/(q-1)}} \Biggl(1 + {e_1(q)\over \log x}+\cdots+{e_k(q)\over \log^k x}+O_k\Bigl({1\over \log^{k+1} x} \Bigr)\Biggr), \] with \(e_j(q)\) being certain constants depending on \(q\), and thus an extension of the main result of the paper. The above remarks (in more detail) along with a study of the numbers \(e_1(q)\), which are closely related to the Euler-Kronecker constant for \({\mathbb K}(q)\), can be found in a recent preprint [``Values of the Euler \(\varphi\)-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields'', Preprint (2011)] of the reviewer with \textit{K. Ford} and \textit{P. Moree}\}.
reviewed by: Florian Luca
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 5526902
zbMATH Keywords: Euler function
zbMATH Keywords: application of L-functions
MaRDI profile type: Publication
full work available at URL: s11512-005-0001-6
OpenAlex ID: W2016638119
cites work: On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions \(\varphi\) and \(\sigma\)
cites work: Q3997223
cites work: A Problem of Rankin on Sums of Powers of Cusp-Form Coefficients
cites work: Mertens' theorem for arithmetic progressions
cites work: Das asymptotische Verhalten von Summen über multiplikative Funktionen
DOI: 10.1007/S11512-005-0001-6
description: scientific article; zbMATH DE number 5526902