Trigonometric analogue of the identities associated with twisted sums of divisor functions
Debika Banerjee, Khyati Khurana
TL;DR
The paper derives trigonometric analogue identities for twisted sums of divisor functions, extending the Cohen-type and Voronoï frameworks to finite trigonometric sums paired with doubly infinite Bessel-series. By leveraging Dirichlet characters, Hurwitz zeta functions, Gauss sums, and the $K$-Bessel machinery, it provides explicit formulas that connect trig sums with the corresponding twisted-sum identities established in prior work (devika2023). A notable application yields a Hardy-type representation for $r_6(n)$, i.e., the number of representations of $n$ as a sum of six squares, in a trig-analytic setting. Overall, the work unifies and extends Ramanujan-type identities within a trigonometric and L-function framework, offering a versatile toolkit for Riesz-sum identities and their character analogues.
Abstract
Inspired by two entries published in Ramanujan's lost notebook on Page 355, B. C. Berndt et al.\cite{MR3351542} presented Riesz sum identities for Ramanujan entries by introducing the twisted divisor sums. Later, S. Kim \cite{MR3541702} derived analogous results by replacing twisted divisor sums with twisted sums of divisor functions. Recently, the authors \cite{devika2023} of the present paper deduced the Cohen-type identities as well as Voronoï summation formulas associated with these twisted sums of divisor functions. The present paper aims to derive an equivalent version of the results in the previous paper in terms of identities involving finite sums of trigonometric functions and the doubly infinite series. As an application, the authors provide an identity for $r_6(n)$, which is analogous to Hardy's famous result where $r_6(n)$ denotes the number of representations of natural number $n$ as a sum of six squares.