Distribution and non-vanishing of special values of \(L\)-series attached to Erdős functions. (English) Zbl 1475.11160
Let \(q\) be a positive integer and \(f\) be a \(q\)-periodic arithmetic function such that \(f(n)\in\{-1,1\}\) when \(q\not|n\) and \(f(n)=0\) otherwise. Erdős conjectured that \(\sum_{n=1}^{\infty}f(n)/n\neq 0\) whenever this series converges (cf. [A. E. Livingston, Can. Math. Bull. 8, 413–432 (1965; Zbl 0129.02801)]). The author shows here that Erdős conjecture holds with “probability” 1. This improves on a result by T. Chatterjee and M. R. Murty [Pac. J. Math. 275, No. 1, 103–113 (2015; Zbl 1333.11084)].
A rational-valued \(q\)-period arithmetic function \(f\) is called an Erdős function \(\mod q\) if \(f(n)\in\{-1,1\}\) when \(q\not|n\) and \(f(n)=0\) otherwise and \(\sum_{a=1}^{q}f(a)=0\). The \(L\)-series associated to \(f\) is \(L(s,f)=\sum_{n=1}^{\infty}f(n)/n^s\). The author also obtains the characteristic function of the limiting distribution of \(L(k,f)\) for any positive integer \(k\) and Erdős function \(f\) with the same parity as \(k\).
A rational-valued \(q\)-period arithmetic function \(f\) is called an Erdős function \(\mod q\) if \(f(n)\in\{-1,1\}\) when \(q\not|n\) and \(f(n)=0\) otherwise and \(\sum_{a=1}^{q}f(a)=0\). The \(L\)-series associated to \(f\) is \(L(s,f)=\sum_{n=1}^{\infty}f(n)/n^s\). The author also obtains the characteristic function of the limiting distribution of \(L(k,f)\) for any positive integer \(k\) and Erdős function \(f\) with the same parity as \(k\).
Reviewer: Jasson Vindas (Gent)
Keywords:
distribution of values of \(L\)-series; non-vanishing of values of \(L\)-series; moments of values of \(L\)-series; Erdős’s conjectureReferences:
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