On a conjecture of Erdős. (English) Zbl 1157.11031

Hessami Pilehrood, T.; Hessami Pilehrood, Kh.

Math. Notes 83, No. 2, 281-284 (2008); translation from Mat. Zametki 83, No. 2, 312-315 (2008).

This paper is concerned with the following conjecture of Erdős: If \(q\) is a positive integer and \(f(x)\) is a number–theoretic function modulo \(q\) for which \(f(n)\in\{-1,1\}\) when \(n=1,2,\ldots,q-1\) and \(f(q)=0\), then \[ S:=\sum_{n\geq 1}\frac{f(n)}{n}\neq 0 \] whenever the series is convergent.
Based on an extensive computer experiment, Tengely [see R. Tijdeman, Bolyai Soc. Math. Stud. 15, 381–405 (2006; Zbl 1103.68103)] showed that the restriction \(f(q)=0\) cannot be removed by giving an explicit function \(f\) with period 36 for which \(S=0\). The authors provide a new proof of Tengely’s result based on a functional relation for the logarithmic derivative of the \(\Gamma\)-function.

Reviewer: Michael Coons (Burnaby, British Columbia)

MSC:

11J81	Transcendence (general theory)
11A25	Arithmetic functions; related numbers; inversion formulas

Keywords:

Erdős conjecture; Euler function; periodic function; gamma function

Citations:

Zbl 1103.68103

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