Erdősian functions and an identity of Gauss

Proceedings of the Japan Academy, Series A. Mathematical Sciences

article

Tapas Chatterjee¹  Suraj Singh Khurana¹ 

[1] Department of Mathematics, Indian Institute of Technology Ropar

关键词: Dirichlet series;    Erd}os conjecture;    Gauss identity;    digamma function.;   

DOI  :  10.3792/pjaa.95.58

学科分类：数学（综合）

来源: Japan Academy

【 摘 要 】

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.

【 授权许可】

Unknown