Divisor sum identities
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 formula_1, or equivalently the Dirichlet convolution of an arithmetic function formula_2 with one: These identities include applications to sums of an arithmetic function over just the proper prime divisors of formula_1. 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 formula_2 to be expressed in  terms of formula_7 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 formula_2 defined as a divisor sum of another arithmetic function formula_7. 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. Average order sum identities. Interchange of summation identities. 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 formula_10 are any prescribed arithmetic functions and that formula_11 denotes the summatory function of formula_7. 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. 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 formula_19 for suitable, application-defined arithmetic functions "u" and "v". A short survey of this method can be found here. Periodic divisor sums. An arithmetic function is "periodic (mod k)", or "k"-periodic, if formula_20 for all formula_21. Particular examples of "k"-periodic number theoretic functions are the Dirichlet characters formula_22 modulo "k" and the greatest  common divisor function formula_23. 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 formula_25 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 Fourier transforms of the GCD. 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 formula_29 using the following result  where formula_30 is a Ramanujan sum (cf. Fourier transform of the totient function): Thus by combining the results above we obtain that Sums over prime divisors. Let the function formula_33 denote the characteristic function of the primes, i.e., formula_34 "if and only if" formula_1 is prime and is zero-valued otherwise. 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 where formula_38 denotes the prime-counting function. Here  we typically make the assumption that the function "f" is continuous and differentiable. Some lesser appreciated divisor sum identities. We have the following divisor sum formulas for "f" any arithmetic function and "g" completely multiplicative where formula_39 is Euler's totient function and formula_40 is the Möbius function: The Dirichlet inverse of an arithmetic function. We adopt the notation that formula_49 denotes the multiplicative identity of Dirichlet convolution so that formula_50 for any arithmetic function "f" and formula_51. The Dirichlet inverse of a function "f" satisfies formula_52 for all formula_51. There is a well-known recursive convolution formula for computing  the Dirichlet inverse formula_54 of a function "f" by induction given in the form of For a fixed function "f", let the function formula_56 Next, define the following two multiple, or nested, convolution variants for any fixed arithmetic function "f": The function formula_58 by the equivalent pair of summation formulas in the next equation is closely related to the Dirichlet inverse for an arbitrary function "f". In particular, we can prove that A table of the values of formula_58 for formula_62 appears below. 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. Let formula_63 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 formula_64 given by