A general class of Voronoi's and Koshliakov–Ramanujan's summation formulas involving d k (n)

Abstract

By using the theory of the Mellin and Mellin convolution type transforms, we prove a general summation formula of Voronoi involving sums of the form ∑ d k (n)f(n), where d k (n), k=2, 3, …, d 2(n)≡ d(n) is the number of ways of expressing n as a product of k factors. These sums are related to the famous Dirichlet divisor problem of determining the asymptotic behaviour as x→∞ of the sum D k (x)=∑ n≤x d k (n). In particular, we generalize Koshliakov's formula and certain identities from Ramanujan's lost notebook to the case of hyper-Bessel functions and Jacobi elliptic theta functions. New examples of Voronoi's summation formulas involving Bessel, exponential functions and their products, which are based on a comprehensive Marichev's table of Mellin's transforms are given. The equivalence of these relations to the functional equation for the Riemann Zeta -function is discussed. An extension of the Koshliakov formula involving the Kontorovich–Lebedev transform is obtained.