User:Virginia-American/Sandbox/divisor convolution

Divisor sum convolutions

edit

The sequence   is called the discrete convolution or the Cauchy product of the sequences an and bn. For integers   and   define the convolution sum  . Note that  

For odd integers  , the sum   can be evaluated in terms of  . Namely:

 
 
 
 
 
 
 
 
 

These are the only   that can be evaluated in terms of divisor sums and polynomials in  . For odd integers  , evaluating the sum requires the Ramanujam function  . For example:

 

There are many other similar formulas. For example:

 





See Eisenstein series for a discussion of the series and functional identities involved in these formulas.[1]

     [2]
     [3]
     [3][4]
     [2][5]
      where τ(n) is Ramanujan's function.    [6][7]

Since σk(n) (for natural number k) and τ(n) are integers, the above formulas can be used to prove congruences[8] for the functions. See Ramanujan tau function for some examples.

Extend the domain of the partition function by setting p(0) = 1.

     [9]   This recurrence can be used to compute p(n).
  1. ^ The paper by Huard, Ou, Spearman, and Williams in the external links also has proofs.
  2. ^ a b Ramanujan, On Certain Arithmetical Functions, Table IV; Papers, p. 146
  3. ^ a b Koblitz, ex. III.2.8
  4. ^ Koblitz, ex. III.2.3
  5. ^ Koblitz, ex. III.2.2
  6. ^ Koblitz, ex. III.2.4
  7. ^ Apostol, Modular Functions ..., Ex. 6.10
  8. ^ Apostol, Modular Functions..., Ch. 6 Ex. 10
  9. ^ G.H. Hardy, S. Ramannujan, Asymptotic Formulæ in Combinatory Analysis, § 1.3; in Ramannujan, Papers p. 279