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

Inspired by two entries published in Ramanujan’s lost notebook on p. 355, Berndt, Kim, and Zaharescu presented Riesz sum identities for the twisted divisor sums. Subsequently, Kim derived analogous results by replacing twisted divisor sums with twisted sums of divisor functions. In a recent work, the authors of the present paper deduced Cohen-type identities and Voronoï summation formulas associated with these twisted sums of divisor functions. The present paper aims to derive an equivalent version of the results offered in the previous article in terms of identities involving finite sums of trigonometric functions and the doubly infinite series. As an application, the authors provide an identity for \(r_6(n)\) $r_6(n)$ , which is analogous to Hardy’s famous result where \(r_6(n)\) $r_6(n)$ denotes the number of representations of natural number n as a sum of six squares.