Proc. Japan Acad. Ser. A Math. Sci. 95 (6), 58-63, (June 2019) DOI: 10.3792/pjaa.95.58
Tapas Chatterjee, Suraj Singh Khurana
KEYWORDS: Dirichlet series, Erdős conjecture, Gauss identity, digamma function, 11M35, 05A19, 33E99
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.