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

We derive Voronoï summation formulas for the Liouville function $\lambda(n)$ , the Möbius 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 the 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.