Transcendence of generalized Euler-Lehmer constants
type: Item
instance of: scholarly article
title: Transcendence of generalized Euler-Lehmer constants
zbMATH Open document ID: 1325.11071
author: Sanoli Gun
author: Ekata Saha
author: Sneh Bala Sinha
published in: Journal of Number Theory
publication date: 2014-09-12T00:00:00Z
review text: The authors study the generalized Euler-Lehmer constants defined as follows: \[ \gamma(\Omega, a,q)= \lim_{x\to+\infty} \Biggl(\Biggl(\sum_{\substack{ n\leq x,\\ n\equiv a\pmod q}} {1_\Omega(n)\over n}\Biggr)- \delta_\Omega{\log(x)\over q}\Biggr), \] where \(\Omega\) be a finite set of primes, \[ P= P_\Omega= \prod_{p\in\Omega} p,\;\delta_\Omega= \prod_{p\in\Omega} \Biggl(1-{1\over p}\Biggr) m,\;\{q,a\}\subset\mathbb N,\;(q,aP)= 1, \] and \(1_\Omega(n)\) is the characteristic function of the set \(\{n\in\mathbb N: (n,P)= 1\}\); clearly, \(\gamma(\emptyset,1,1)\) coincides with the famous Euler constant \(\gamma\), and \(\gamma(\Omega)= \gamma(\Omega,1,1)\) is the Lehmer constant. The goal of the authors is to prove the following assertions: Theorem 1. Let \(q>1\). Then the number \(\gamma(\Omega,a,q)- \delta(\Omega)\gamma/q\) is transcendental. Theorem 2. Let \(a\) and \(q>1\) be natural numbers with \((a,q)= 1\) and \(S\) be the set of prime divisors of \(q\). Also let \[ U= \{\Omega:\Omega\text{ is finite set of primes such that }\Omega\cap S=\emptyset\}. \] Then the set \(T= \{\gamma(\Omega,a,q): \Omega\in U\}\) is infinite and can contain no more than one algebraic element. Theorem 3. Let \(a\in\mathbb N\), \(\Omega\) be a finite set of primes, and \(P=\prod_{p\in\Omega}\). Let further \(S= \{q_1,q_2,\dots\}\) be an infinite set of mutually coprime natural numbers \(q_i>1\) for all \(i\in\mathbb N\). Further suppose that all \(q_i\) is coprime to \(aP\). Then the set \[ T= \{\gamma(\Omega, a,q_i): q_i\in S\} \] can contain no more than one algebraic element. The authors represent \(\gamma(\Omega,a,q)\) in the form \[ \gamma(\Omega,a,q)={1\over \varphi(q)} \sum_{\substack{ \chi\text{\,mod\,}q\\ \chi\neq\chi_0}} \overline\chi(a) L(1,\chi) \prod_{p\in\Omega} \Biggl(1-{\chi(p)\over p}\Biggr)+{\delta_\Omega\over q} \Biggl(\gamma+ \sum_{p|Pq} {\log(p)\over p-1}\Biggr). \] Then they take in account that 1. for odd \(\chi\) the number \(L(1,\chi)\) is product of \(\pi\)…
reviewed by: Leonid A. Gutnik
Mathematics Subject Classification ID: 11J81
zbMATH DE Number: 6343988
zbMATH Keywords: generalized Euler-Lehmer constants
zbMATH Keywords: Baker's theory of linear forms in logarithms
MaRDI profile type: Publication
full work available at URL: j.jnt.2014.06.010
OpenAlex ID: W2042403539
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DOI: 10.1016/J.JNT.2014.06.010
description: scientific article; zbMATH DE number 6343988