## Abstract

We prove an explicit integral formula for computing the product of two shifted Riemann zeta functions everywhere in the complex plane. We show that this formula implies the existence of infinite families of exact exponential sum identities involving the divisor functions, and we provide examples of these identities. We conjecturally propose a method to compute divisor functions by matrix inversion, without employing arithmetic techniques.

Original language | English (US) |
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Article number | 128827 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 542 |

Issue number | 2 |

DOIs | |

State | Published - Feb 15 2025 |

### Funding

B.S. would like to acknowledge the Northwestern University Amplitudes and Insights Group, Department of Physics and Astronomy, and Weinberg College for support. The work of B.S. was supported in part by the Department of Energy under Award Number DE-SC0021485. N. R. wishes to acknowledge support from Vikram Kilambi and Chris Pernin. The authors are sincerely grateful to the referee for making valuable suggestions and corrections that have greatly increased the quality of the paper. B. S. would like to acknowledge the Northwestern University Amplitudes and Insight group, Department of Physics and Astronomy, and Weinberg College for support. The work of B. S. was supported in part by the Department of Energy under Award Number DE-SC0021485. The authors are sincerely grateful to the referee for making valuable suggestions and corrections that have greatly increased the quality of the paper.

## Keywords

- Exact exponential sums involving arithmetic functions
- Generalized divisor functions
- Matrix techniques for equation solving
- Riemann zeta function
- Special functions

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics