Dirichlet convolution

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In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.

DefinitionEdit

If   are two arithmetic functions from the positive integers to the complex numbers, the Dirichlet convolution fg is a new arithmetic function defined by:

 

where the sum extends over all positive divisors d of n, or equivalently over all distinct pairs (a, b) of positive integers whose product is n.

This product occurs naturally in the study of Dirichlet series such as the Riemann zeta function. It describes the multiplication of two Dirichlet series in terms of their coefficients:

 

PropertiesEdit

The set of arithmetic functions forms a commutative ring, the Dirichlet ring, under pointwise addition, where f + g is defined by (f + g)(n) = f(n) + g(n), and Dirichlet convolution. The multiplicative identity is the unit function ε defined by ε(n) = 1 if n = 1 and ε(n) = 0 if n > 1. The units (invertible elements) of this ring are the arithmetic functions f with f(1) ≠ 0.

Specifically, Dirichlet convolution is[1] associative,

 

distributes over addition

 ,

is commutative,

 ,

and has an identity element,

  =  .

Furthermore, for each   having  , there exists an arithmetic function   with  , called the Dirichlet inverse of  .

The Dirichlet convolution of two multiplicative functions is again multiplicative, and every not constantly zero multiplicative function has a Dirichlet inverse which is also multiplicative. In other words, multiplicative functions form a subgroup of the group of invertible elements of the Dirichlet ring. Beware however that the sum of two multiplicative functions is not multiplicative (since  ), so the subset of multiplicative functions is not a subring of the Dirichlet ring. The article on multiplicative functions lists several convolution relations among important multiplicative functions.

Another operation on arithmetic functions is pointwise multiplication: fg is defined by (fg)(n) = f(n) g(n). Given a completely multiplicative function  , pointwise multiplication by   distributes over Dirichlet convolution:  .[2] The convolution of two completely multiplicative functions is multiplicative, but not necessarily completely multiplicative.

ExamplesEdit

In these formulas, we use the following arithmetical functions:

  •   is the multiplicative identity:  , otherwise 0.
  •   is the constant function with value 1:   for all  . Keep in mind that   is not the identity. (Some authors denote this as   because the associated Dirichlet series is the Riemann zeta function.)
  •   for   is a set indicator function:   iff  , otherwise 0.
  •   is the identity function with value n:  .
  •  is the kth power function:  .

The following relations hold:

  •  , the Dirichlet inverse of the constant function   is the Möbius function. Hence:
  •   if and only if  , the Möbius inversion formula
  •  , the kth-power-of-divisors sum function σk
  •  , the sum-of-divisors function σ = σ1
  •   , the number-of-divisors function d(n) = σ0
  •  ,  by Möbius inversion of the formulas for σk, σ, and d
  •  
  •  
  •   , proved under Euler's totient function
  •   , by Möbius inversion
  •    , from convolving 1 on both sides of  
  •    where λ is Liouville's function
  •   where Sq = {1, 4, 9, ...} is the set of squares
  •  
  •  
  •  , Jordan's totient function
  •  
  •  , where   is von Mangoldt's function
  •   where   is the prime omega function counting distinct prime factors of n
  •   is an indicator function where the set   is the collection of positive primes and integral powers of two.
  •   where   is the characteristic function of the primes.

This last identity shows that the prime counting function is given by the summatory function

 

where   is the Mertens function and   is the distinct prime factor counting function from above. This expansion follows from the identity for the sums over Dirichlet convolutions given on the divisor sum identities page (a standard trick for these sums)[3].

Dirichlet inverseEdit

ExamplesEdit

Given an arithmetic function   its Dirichlet inverse   may be calculated recursively: the value of   is in terms of   for  .

For  :

 , so
 . This implies that   does not have a Dirichlet inverse if  .

For  :

 ,
 ,

For  :

 ,
 ,

For  :

 ,
 ,

and in general for  ,

 

PropertiesEdit

The following properties of the Dirichlet inverse hold:[4]

  • The function f has a Dirichlet inverse if and only if f(1) ≠ 0.
  • The Dirichlet inverse of a multiplicative function is again multiplicative.
  • The Dirichlet inverse of a Dirichlet convolution is the convolution of the inverses of each function:  .
  • A multiplicative function f is completely multiplicative if and only if  .
  • If f is completely multiplicative then   whenever   and where   denotes pointwise multiplication of functions.

Other formulasEdit

Arithmetic function Dirichlet inverse:[5]
Constant function with value 1 Möbius function μ
   
Liouville's function λ Absolute value of Möbius function |μ|
Euler's totient function    
The generalized sum-of-divisors function    

An exact, non-recursive formula for the Dirichlet inverse of any arithmetic function f is given in Divisor sum identities. A more partition theoretic expression for the Dirichlet inverse of f is given by

 

Dirichlet seriesEdit

If f is an arithmetic function, one defines its Dirichlet series generating function by

 

for those complex arguments s for which the series converges (if there are any). The multiplication of Dirichlet series is compatible with Dirichlet convolution in the following sense:

 

for all s for which both series of the left hand side converge, one of them at least converging absolutely (note that simple convergence of both series of the left hand side DOES NOT imply convergence of the right hand side!). This is akin to the convolution theorem if one thinks of Dirichlet series as a Fourier transform.

Related conceptsEdit

The restriction of the divisors in the convolution to unitary, bi-unitary or infinitary divisors defines similar commutative operations which share many features with the Dirichlet convolution (existence of a Möbius inversion, persistence of multiplicativity, definitions of totients, Euler-type product formulas over associated primes, etc.).

Dirichlet convolution is the convolution of the incidence algebra for the positive integers ordered by divisibility.

See alsoEdit

ReferencesEdit

  1. ^ Proofs of all these facts are in Chan, ch. 2
  2. ^ A proof is in the article Completely multiplicative function#Proof of distributive property.
  3. ^ Schmidt, Maxie. Apostol's Introduction to Analytic Number Theory. This identity is a little special something I call "croutons". It follows from several chapters worth of exercises in Apostol's classic book.
  4. ^ Again see Apostol Chapter 2 and the exercises at the end of the chapter.
  5. ^ See Apostol Chapter 2.

External linksEdit