Proceedings of the Japan Academy, Series A, Mathematical Sciences

Erdősian functions and an identity of Gauss

Tapas Chatterjee and Suraj Singh Khurana

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Abstract

A famous identity of Gauss gives a closed form expression for the values of the digamma function $\psi(x)$ at rational arguments $x$ in terms of elementary functions. Linear combinations of such values are intimately connected with a conjecture of Erdős which asserts non vanishing of an infinite series associated to a certain class of periodic arithmetic functions. In this note we give a different proof for the identity of Gauss using an orthogonality like relation satisfied by these functions. As a by product we are able to give a new interpretation for $n$th Catalan number in terms of these functions.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 95, Number 6 (2019), 58-63.

Dates
First available in Project Euclid: 31 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.pja/1559268171

Digital Object Identifier
doi:10.3792/pjaa.95.58

Mathematical Reviews number (MathSciNet)
MR3960282

Subjects
Primary: 11M35: Hurwitz and Lerch zeta functions 05A19: Combinatorial identities, bijective combinatorics 33E99: None of the above, but in this section

Keywords
Dirichlet series Erdős conjecture Gauss identity digamma function

Citation

Chatterjee, Tapas; Khurana, Suraj Singh. Erdősian functions and an identity of Gauss. Proc. Japan Acad. Ser. A Math. Sci. 95 (2019), no. 6, 58--63. doi:10.3792/pjaa.95.58. https://projecteuclid.org/euclid.pja/1559268171


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