generalized ramanujan sum
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2016 ◽  
Vol 12 (02) ◽  
pp. 383-408 ◽  
Author(s):  
Patrick Kühn ◽  
Nicolas Robles

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.


2013 ◽  
Vol 13 (02) ◽  
pp. 1350085 ◽  
Author(s):  
YOUSEF ZAMANI ◽  
ESMAEIL BABAEI

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.


Author(s):  
Vichian Laohakosol ◽  
Pattira Ruengsinsub ◽  
Nittiya Pabhapote

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.


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