Factorization theorems for relatively prime divisor sums, GCD sums and generalized Ramanujan sums
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Factorization theorems for relatively prime divisor sums, GCD sums and generalized Ramanujan sums Hamed Mousavi1 · Maxie D. Schmidt1 Received: 1 March 2019 / Accepted: 14 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We build on and generalize recent work on so-termed factorization theorems for Lambert series generating functions. These factorization theorems allow us to express formal generating functions for special sums as invertible matrix transformations involving partition functions. In the Lambert series case, the generating functions at hand enumerate the divisor sum coefficients of q n as d|n f (d) for some arithmetic function f . Our new factorization theorems provide analogs to these established expansions generating corresponding sums of the form d:(d,n)=1 f (d) (type I sums) and the Anderson–Apostol sums d|(m,n) f (d)g(n/d) (type II sums) for any arithmetic functions f and g. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function p(n) to arbitrary arithmetic functions f . We conclude the last section of the article by directly expanding new formulas for an arithmetic function g by the type II sums using discrete, and discrete time, Fourier transforms (DFT and DTFT) for functions over inputs of greatest common divisors. Keywords Divisor sum · Totient function · Matrix factorization · Möbius inversion · Partition function Mathematics Subject Classification 11N64 · 11A25 · 05A17
1 Notation and conventions To make a common source of references to definitions of key functions and sequences defined within the article, we provide a comprehensive list of commonly used nota-
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Maxie D. Schmidt [email protected]; [email protected] Hamed Mousavi [email protected]
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School of Mathematics, Georgia Institute of Technology, 117 Skiles Building, 686 Cherry Street NW, Atlanta, GA 30332, USA
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H. Mousavi, M. D. Schmidt
tion and conventions. Where possible, we have included section references to local definitions of special sequences and functions where they are given in the text. We have organized the symbols list alphabetically by symbol. This listing of notation and conventions is a useful reference to accompany the article. It is provided in Appendix A.
2 Introduction 2.1 Motivation The average order of an arithmetic function f is defined as the arithmetic mean of the summatory function, F(x) := n≤x f (n), as F(x)/x. We typically represent the average order of a function in the form of an asymptotic formula in the cases where the formula, F(x)/x, for the average order diverges as x → ∞. For example, it is well known that the average order of (n), which counts the number of prime factors of n (counting multiplicity), is log log n, and that the average order of Euler’s totient function, φ(n), is given by π6n2 [4]. We are motivated by considering the breakdown of the partial sums of an arithmetic function f (d) whose average order we would like to estimate into sums over the pairwise disjoint se
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