Arithmetical Inversion Formulas

Abstract

Let n and r be integers, r positive, and define the coreγ(r) of r to be the product of the distinct prime factors of r (γ(1) = 1). Let f(n,r) be a complex-valued, arithmetical function of n and r. If for all n,f﹛n,r) = f((n,r), r) then f(n, r) is called an even function (mod r), and if f(n,r) = f(γ(n, r), r) for all n, γ(n, r) = γ((n, r)), then f(n, r) is said to be a primitive function (mod r). Clearly, both classes of functions are subclasses of the periodic functions (mod r), while the primitive functions form a subclass of the even functions (mod r).In a series of three papers (3; 5; 6) the author developed parallel, though interrelated, trigonometric and arithmetical theories of the even and primitive functions (mod r).