Voronoi summation formulas, oscillations of Riesz sums, and Ramanujan-Guinand and Cohen type identities

TL;DR

This paper develops advanced summation formulas for important arithmetic functions, explores their connections to classical identities, and investigates the oscillatory behavior of related sums under key hypotheses like the Riemann Hypothesis.

Contribution

It derives new Voronoi summation formulas for the Liouville, Möbius, and divisor functions, linking them to zeros of the Riemann zeta function and classical identities.

Findings

Derived Voronoi formulas involving zeros of zeta function.

Established Cohen and Ramanujan-Guinand type identities.

Analyzed oscillations of Riesz sums under RH and related conjectures.

Abstract

We derive Vorono\"{\dotlessi} summation formulas for the Liouville function $\lambda(n)$, the M\"{o}bius function $\mu(n)$, and for $d^{2}(n)$, where $d(n)$ is the divisor function. The formula for $\lambda(n)$ requires explicit evaluation of certain infinite series for which the use of the Vinogradov-Korobov zero-free region of the Riemann zeta function is indispensable. Several results of independent interest are obtained as special cases of these formulas. For example, a special case of the one for $\mu(n)$ is a famous result of Ramanujan, Hardy, and Littlewood. Cohen type and Ramanujan-Guinand type identities are established for $\lambda(n)$ and $\sigma_a(n)\sigma_b(n)$, where $\sigma_s(n)$ is the generalized divisor function. As expected, infinite series over the non-trivial zeros of $\zeta(s)$ now form an essential part of all of these formulas. A series involving $\sigma_a(n)\sigma_b(n)$ and product of modified Bessel functions occurring in one of our identities has appeared in a recent work of Dorigoni and Treilis in string theory. Lastly, we obtain results on oscillations of Riesz sums associated to $\lambda(n), \mu(n)$ and of the error term of Riesz sum of $d^2(n)$ under the assumption of the Riemann Hypothesis, simplicity of the zeros of $\zeta(s)$, the Linear Independence conjecture, and a weaker form of the Gonek-Hejhal conjecture.