Transcendence of generalized Euler-Lehmer constants
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 \(\pi\) and algebraic number; 2. for even \(\chi\) the number \(L(1,\chi)\) is a nonzero algebraic multiple of \[ \sum_{\substack{ 1<a<q/2\\ (a,q)= 1}} \log(\xi_a), \] where \(\xi_a\) are Ramachandra's units, and (as units) these units are multiplicatively independent with any finite set of mutually distinct prime \(p\). After that, the authors deduce their assertions from known results of \textit{A. Baker} [Transcendental number theory. Paperback ed. Cambridge etc.: Cambridge University Press (1990; Zbl 0715.11032)], \textit{M. Ram Murty} and \textit{N. Saradha} [Acta Arith. 133, No. 4, 349--362 (2008; Zbl 1253.11077); J. Number Theory 130, No. 12, 2671--2682 (2010; Zbl 1204.11114)] and \textit{M. Ram Murty} and \textit{V. Kumar Murty} [J. Number Theory 131, No. 9, 1723--1733 (2011; Zbl 1241.11083)]. Reviewer's remarks: {\parindent=6mm \begin{itemize}\item[1.] The authors refer to Theorem 2.2 of Baker's book. But this theorem is a corollary of Baker's main Theorem 2.1. The article would be shorter, if the authors used Theorem 2.1 instead of Theorem 2.2. \item[2.] The assertion that the set, which consists of (finite number of mutually different) primes and (finite number) of multiplicatively independent units, is multiplicatively independent set, can be left to the reader, as easy exercise; but one can meet the proof of this evident assertion twice in the text, in Proposition 9 and Proposition 11. \end{itemize}}
Baker's theory of linear forms in logarithms
scientific article; zbMATH DE number 6343988