Divisor sum identities
From Wikipedia, the free encyclopedia
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:
![]() | This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)
|
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.