Further Generalization of Ramanujan Sums with Regular A-Functions

TL;DR

This work extends Ramanujan sums by introducing a two-parameter framework $C_{(A_1,A_2)}(m,n)$ built from regular $A$-functions, unifying and generalizing several known sums. It establishes a distributive, complete lattice structure on the regular $A$-functions, and shows that the Hölder-type identity $C_{(A_1,A_2)}(m,n)=\Phi_{(A_1,A_2)}(m,n)$ holds iff $A_2\le A_1$, with a corresponding generalized von Sterneck function $\Phi_{(A_1,A_2)}$. The paper also derives an orthogonality relation, a trigonometric-sum representation valid only in the classical $A_1=D$ case, and a comprehensive theory for multivariable Ramanujan expansions $f(n_1,\dots,n_k)=\sum a(q_1,...,q_k)\prod_{j=1}^k C_{(A_1,A_2)}(n_j,q_j)$, enabling expansions for arbitrary regular $A$-functions. Overall, it provides a unified, extensible framework for analyzing arithmetic functions via regular $A$-function convolutions, with new expansions and identities that generalize and extend prior work on Ramanujan sums and their multivariable counterparts.

Abstract

In the study of Ramanujan sums, the so-called regular $A$-function is a set-valued multiplicative function that tracks certain subsets of the divisor sets of natural numbers. McCarthy provided a generalization of the Ramanujan sum using these regular $A$-function based arithmetic convolutions. This approach has recently attracted considerable interest from several researchers. In this paper, we extend McCarthy's generalization by introducing two regular $A$-functions corresponding to both parameters in the Ramanujan sum. Fortunately, these sums exhibit several properties of the Ramanujan sums. We also generalize the greatest common divisor (GCD) function and the Von Sterneck formula. Our introduction of two regular $A$-functions into these expressions enables us to explore a novel perspective on the connection between these expressions and the order relation between the two regular $A$-functions. In particular, we establish the necessary and sufficient conditions for orthogonality and Dedekind-Hölder's identity (i.e., Ramanujan sum = Von Sterneck function) to hold. Our primary motivation for this further generalization proposed in this paper is expansions of arithmetic functions based on arbitrary regular $A$-functions. To the best of our knowledge, the expansions of arbitrary $A$-functions discussed here are new in the literature.

Figures (1)

• Figure 1: A depiction of the poset of regular $A$-functions. Here $D$ denotes a complete set of divisors and $U$ is the set of unitary divisors.

Theorems & Definitions (49)

• Theorem : Generalized Dedekind-Hölder Identity
• Definition 1.0.1
• Theorem 1.0.1
• Definition 1.0.2: Generalization of the Ramanujan sum
• Theorem 1.0.2: Multiplicativity
• proof : Proof.
• Corollary 1.0.3: Evaluations at prime powers
• proof : Proof.
• Theorem 1.0.4: Sum over regular divisors
• proof : Proof.