Abstract
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.
Highlights
Introduction and basic definitions The classicalRamanujan sum is the arithmetic function of two variables c(n, k) =e2π imn/k, m(mod k) gcd(m,k)=1 (1.1)where k ∈ N, n ∈ N0 := N ∪{0}
The second is a derivation of the mean value of a generalized Ramanujan sum (GRS)
Introduction and basic definitions The classical Ramanujan sum is the arithmetic function of two variables
Summary
We derive many more properties about GRSs, similar to those in the beginning of [15, Chapter 2] and [2, Chapter 8]. Let α ∈ C and n, r ∈ N with prime factorizations n = pa, r = pb, where a, b are nonnegative integers. The following corollary, known as Holder relation, is [2, Theorem 8.8] and [15, Theorem 2.3] It is a special case of a more general result in [1, 6], so we omit its proof. Let n,r be a unitary pair and N = r/ gcd(n,r). Taking α = β = 1 in Theorem 2.7, we have g −1 c(a, s)c(b, t) = r μ0 s1 gcd(k, g) μ0 t1 gcd(k, g) e2πikn/g , a+b≡n(mod r ). For a multiplicative function f , define Fm(n) = f (mn)/ f (m) (n, m ∈ N) It is proved in [18] that. If f is a totient [13], that is, a Dirichlet product of a completely multiplicative function with an inverse of a completely multiplicative function,
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