An arithmetic Poisson formula

Let α(n) denote any arithmetic function. Since most arithmetic functions that are encountered are defined on only the nonnegative integers, we define α(−n) = α(n) for each positive integer n. Our objective is to develop a Poisson type formula for Σ^∞_n=−∞α(n)f(n), where f belongs to a suitable class of functions. We conclude the paper with several applications of our arithmetic Poisson formula.