Explicit formulas of a generalized Ramanujan sum

In this paper, explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.

THE DIMENSIONS OF CYCLIC SYMMETRY CLASSES OF POLYNOMIALS

The dimensions of the symmetry classes of polynomials with respect to a certain cyclic subgroup of Sm generated by an m-cycle are explicitly given in terms of the generalized Ramanujan sum. These dimensions can also be expressed in terms of the Euler ϕ-function and the Möbius function for some special cases.

Ramanujan sums via generalized Möbius functions and applications

A generalized Ramanujan sum (GRS) is defined by replacing the usual Möbius function in the classical Ramanujan sum with the Souriau-Hsu-Möbius function. After collecting basic properties of a GRS, mostly containing existing ones, seven aspects of a GRS are studied. The first shows that the unique representation of even functions with respect to GRSs is possible. The second is a derivation of the mean value of a GRS. The third establishes analogues of the remarkable Ramanujan's formulae connecting divisor functions with Ramanujan sums. The fourth gives a formula for the inverse of a GRS. The fifth is an analysis showing when a reciprocity law exists. The sixth treats the problem of dependence. Finally, some characterizations of completely multiplicative function using GRSs are obtained and a connection of a GRS with the number of solutions of certain congruences is indicated.