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

Abstract

We derive Vorono\"{\dotlessi} summation formulas for the Liouville function λ(n), the Möbius function μ(n), and for d2(n), where d(n) is the divisor function. The formula for λ(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 μ(n) is a famous result of Ramanujan, Hardy, and Littlewood. Cohen type and Ramanujan-Guinand type identities are established for λ(n) and σa(n)σb(n), where σs(n) is the generalized divisor function. As expected, infinite series over the non-trivial zeros of ζ(s) now form an essential part of all of these formulas. A series involving σa(n)σb(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 λ(n),μ(n) and of the error term of Riesz sum of d2(n) under the assumption of the Riemann Hypothesis, simplicity of the zeros of ζ(s), the Linear Independence conjecture, and a weaker form of the Gonek-Hejhal conjecture.