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 , 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.

Average order sum identities

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Interchange of summation identities

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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.[1]

  1.  
  2.  
  3.  
  4.  
  5.  

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

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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.

Periodic divisor sums

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An arithmetic function is periodic (mod k), or k-periodic, if   for all  . 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]

 

Fourier transforms of the GCD

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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   using the following result where   is a Ramanujan sum (cf. Fourier transform of the totient function):[3]

 

Thus by combining the results above we obtain that

 

Sums over prime divisors

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Let the function   denote the characteristic function of the primes, i.e.,   if and only if   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 [4]

 

where   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

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We have the following divisor sum formulas for f any arithmetic function and g completely multiplicative where   is Euler's totient function and   is the Möbius function:[5][6]

  1.  
  2.  
  3.  
  4. If f is completely multiplicative then the pointwise multiplication   with a Dirichlet convolution yields  .
  5.  
  6. If   and n has more than m distinct prime factors, then  

The Dirichlet inverse of an arithmetic function

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We adopt the notation that   denotes the multiplicative identity of Dirichlet convolution so that   for any arithmetic function f and  . The Dirichlet inverse of a function f satisfies   for all  . There is a well-known recursive convolution formula for computing the Dirichlet inverse   of a function f by induction given in the form of [7]

 

For a fixed function f, let the function  

Next, define the following two multiple, or nested, convolution variants for any fixed arithmetic function f:

 

The function   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   for   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.

n   n   n  
2   7   12  
3   8   13  
4   9   14  
5   10   15  
6   11   16  

Let   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   given by [8]

 

Variants of sums over arithmetic functions

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See also

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Notes

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  1. ^ See also Section 3.10 of Apostol.
  2. ^ Section 27.10 in the NIST Handbook of Mathematical Functions (DLMF).
  3. ^ Schramm, W. (2008). "The Fourier transform of functions of the greatest common divisors". Integers. 8.
  4. ^ See Section 2.2 in Villarino, M. B. (2005). "Mertens' Proof of Mertens' Theorem". arXiv:math/0504289.
  5. ^ In respective order from Apostol's book: Exercise 2.29, Theorem 2.18, and Exercises 2.31-2.32
  6. ^ The first identity has a well-known Dirichlet series of the form   catalogued in Gould, Henry W.; Shonhiwa, Temba (2008). "A catalogue of interesting Dirichlet series". Miss. J. Math. Sci. 20 (1). Archived from the original on 2011-10-02.
  7. ^ See Section 2.7 of Apostol's book for a proof.
  8. ^ a b M. Merca and M. D. Schmidt (2017). "Factorization Theorems for Generalized Lambert Series and Applications". pp. 13–20. arXiv:1712.00611 [math.NT].
  9. ^ This identity is proved in an unpublished manuscript by M. D. Schmidt which will appear on ArXiv in 2018.

References

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