Trigonometric analogue of the identities associated with twisted sums of divisor functions

TL;DR

This paper develops trigonometric identities related to twisted divisor sums, extending classical results and providing new formulas, including an identity for the number of representations of integers as sums of six squares.

Contribution

It derives a trigonometric analogue of identities involving twisted divisor sums and extends previous work by connecting these identities to sums of six squares.

Findings

Derived identities involving finite sums of trigonometric functions and infinite series.

Established an identity for the representation of numbers as sums of six squares.

Extended classical divisor sum identities to a trigonometric framework.

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 Vorono\"i 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.