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Concept
Divisor sum identities
Summary
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number , or equivalently the Dirichlet convolution of an arithmetic function with one:
These identities include applications to sums of an arithmetic function over just the proper prime divisors of .
We also define periodic variants of these divisor sums with respect to the greatest common divisor function in the form of
Well-known inversion relations that allow the function to be expressed in terms of are provided by the Möbius inversion formula.
Naturally, some of the most interesting examples of such identities result when considering the average order summatory functions over an arithmetic function defined as a divisor sum of another arithmetic function . Particular examples of divisor sums involving special arithmetic functions and special Dirichlet convolutions of arithmetic functions can be found on the following pages:
here, here, here, here, and here.
The following identities are the primary motivation for creating this topics page. These identities do not appear to be well-known, or at least well-documented, and are extremely useful tools to have at hand in some applications. In what follows, we consider that are any prescribed arithmetic functions and that denotes the summatory function of . A more common special case of the first summation below is referenced here.
In general, these identities are collected from the so-called "rarities and b-sides" of both well established and semi-obscure analytic number theory notes and techniques and the papers and work of the contributors. The identities themselves are not difficult to prove and are an exercise in standard manipulations of series inversion and divisor sums. Therefore, we omit their proofs here.
The convolution method is a general technique for estimating average order sums of the form
where the multiplicative function f can be written as a convolution of the form for suitable, application-defined arithmetic functions u and v.
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