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

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Abstract

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 \(r_6(n)\), 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.

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Acknowledgements

The authors are grateful to the referee for several helpful suggestions and comments.

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  1. Department of Mathematics, Indraprastha Institute of Information Technology IIIT, Delhi, Okhla, Phase III, New Delhi, 110020, India

    Debika Banerjee & Khyati Khurana

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  1. Debika Banerjee
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  2. Khyati Khurana
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Correspondence to Khyati Khurana.

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Research of the second author was supported by the University Grants Commission, Department of Higher Education, Government of India, under NTA Ref. no. 191620205105.

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Banerjee, D., Khurana, K. Trigonometric analogues of the identities associated with twisted sums of divisor functions. Ramanujan J (2024). https://doi.org/10.1007/s11139-024-00833-6

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