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
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.
A more common special case of the first summation below is referenced here.
[1] 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.
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 g and h. A short survey of this method can be found here.
A related technique is the use of the formula this is known as the Dirichlet hyperbola method.
An arithmetic function is periodic (mod k), or k-periodic, if
Particular examples of k-periodic number theoretic functions are the Dirichlet characters
modulo k and the greatest common divisor function
It is known that every k-periodic arithmetic function has a representation as a finite discrete Fourier series of the form where the Fourier coefficients
defined by the following equation are also k-periodic: We are interested in the following k-periodic divisor sums: It is a fact that the Fourier coefficients of these divisor sum variants are given by the formula [2] We can also express the Fourier coefficients in the equation immediately above in terms of the Fourier transform of any function h at the input of
Then as a special case of the first identity in equation (1) in section interchange of summation identities above, we can express the average order sums We also have an integral formula based on Abel summation for sums of the form [4] where
Here we typically make the assumption that the function f is continuous and differentiable.
We have the following divisor sum formulas for f any arithmetic function and g completely multiplicative where
denotes the multiplicative identity of Dirichlet convolution so that
There is a well-known recursive convolution formula for computing the Dirichlet inverse
by the equivalent pair of summation formulas in the next equation is closely related to the Dirichlet inverse for an arbitrary function f.[8] In particular, we can prove that [9] A table of the values of
This table makes precise the intended meaning and interpretation of this function as the signed sum of all possible multiple k-convolutions of the function f with itself.
where p is the Partition function (number theory).
Then there is another expression for the Dirichlet inverse given in terms of the functions above and the coefficients of the q-Pochhammer symbol for