TY - JOUR
T1 - The Riemann zeta function and exact exponential sum identities of divisor functions
AU - Nastasescu, Maria
AU - Robles, Nicolas
AU - Stoica, Bogdan
AU - Zaharescu, Alexandru
N1 - 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.
PY - 2025/2/15
Y1 - 2025/2/15
N2 - 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.
AB - 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.
KW - Exact exponential sums involving arithmetic functions
KW - Generalized divisor functions
KW - Matrix techniques for equation solving
KW - Riemann zeta function
KW - Special functions
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U2 - 10.1016/j.jmaa.2024.128827
DO - 10.1016/j.jmaa.2024.128827
M3 - Article
AN - SCOPUS:85204190777
SN - 0022-247X
VL - 542
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 2
M1 - 128827
ER -