Identities involving the coefficients of a class of Dirichlet series. VII

Abstract

Let a ( n ) a(n) be an arithmetical function, and consider the Riesz sum A ρ ( x ) = Σ n ≤ x a ( n ) ( x − n ) ρ {A_\rho }(x) = {\Sigma _{n \leq x}}a(n){(x - n)^\rho } . For a ( n ) a(n) belonging to a certain class of arithmetical functions, A ρ ( x ) {A_\rho }(x) can be expressed in terms of an infinite series of Bessel functions. K. Chandrasekharan and R. Narasimhan have established this identity for the widest known range of ρ \rho . Their proof depends upon equi-convergence theory of trigonometric series. An alternate proof is given here which uses only the classical theory of Bessel functions.