Editing Dirichlet convolution

{{Short description|Mathematical operation on arithmetical functions}}
{{More footnotes|date=June 2018}}

[[File:Dirichlet.jpg | thumb | right | Johann Peter Gustav Lejeune Dirichlet]]
In [[mathematics]], '''Dirichlet convolution''' (or '''divisor convolution''') is a [[binary operation]] defined for [[arithmetic function]]s; it is important in [[number theory]]. It was developed by [[Peter Gustav Lejeune Dirichlet]].

==Definition==
If <math>f , g : \mathbb{N}\to\mathbb{C}</math> are two [[Arithmetic function|arithmetic functions]], their Dirichlet convolution <math>f*g</math> is a new arithmetic function defined by:

:<math>
(f*g)(n) \ =\ \sum_{d\,\mid \,n} f(d)\,g\!\left(\frac{n}{d}\right) \ =\  \sum_{ab\,=\,n}\!f(a)\,g(b),
</math>

where the sum extends over all positive [[divisor]]s <math>d</math> of <math>n</math>, or equivalently over all distinct pairs <math>(a,b)</math> of positive integers whose product is <math>n</math>.

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:
:<math>\left(\sum_{n\geq 1}\frac{f(n)}{n^s}\right)
\left(\sum_{n\geq 1}\frac{g(n)}{n^s}\right)
\ = \ 
\left(\sum_{n\geq 1}\frac{(f*g)(n)}{n^s}\right).
</math>

==Properties==

The set of arithmetic functions forms a [[commutative ring]], the '''{{visible anchor|Dirichlet ring}}''', with addition given by [[pointwise addition]] and multiplication by Dirichlet convolution. The multiplicative identity is the [[unit function]] <math>\varepsilon</math> defined by <math>\varepsilon(n)=1</math> if <math>n=1</math> and <math>0</math> otherwise. The [[unit (ring theory)|unit]]s (invertible elements) of this ring are the arithmetic functions <math>f</math> with <math>f(1) \neq 0</math>.

Specifically, Dirichlet convolution is [[associativity|associative]],<ref>Proofs are in Chan, ch. 2</ref>
:<math>(f * g) * h = f * (g * h),</math>
[[distributivity|distributive]] over addition
:<math>f * (g + h) = f * g + f * h</math>,
[[commutativity|commutative]],
:<math>f * g = g * f</math>,
and has an identity element,
: <math>f * \varepsilon</math> = <math>\varepsilon * f = f</math>.
Furthermore, for each function <math>f</math> having <math>f(1) \neq 0</math>, there exists another arithmetic function <math>f^{-1}</math> satisfying <math>f * f^{-1} = \varepsilon</math>, called the '''{{visible anchor|Dirichlet inverse}}''' of <math>f</math>.

The Dirichlet convolution of two [[multiplicative function]]s 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 <math>(f+g)(1)=f(1)+g(1)=2 \neq 1</math>), 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:  <math>fg</math> is defined by <math>(fg)(n)=f(n)g(n)</math>. Given a [[completely multiplicative function]] <math>h</math>, pointwise multiplication by <math>h</math> distributes over Dirichlet convolution: <math>(f * g)h = (fh) * (gh)</math>.<ref>A proof can be found in [[Completely multiplicative function#Proof of distributive property|this article]].</ref> The convolution of two completely multiplicative functions is multiplicative, but not necessarily completely multiplicative.

==Properties and examples==

In these formulas, we use the following [[arithmetical function]]s:
* <math>\varepsilon</math> is the multiplicative identity: <math>\varepsilon(1) = 1</math>, otherwise 0 (<math>\varepsilon(n)=\lfloor \tfrac1n \rfloor</math>).
* <math>1</math> is the constant function with value 1:  <math>1(n) = 1</math> for all <math>n</math>. Keep in mind that <math>1</math> is not the identity. (Some authors [[Incidence algebra#Special_elements|denote this]] as <math>\zeta</math> because the associated Dirichlet series is the [[Riemann zeta function]].)
* <math>1_C</math> for <math>C \subset \mathbb{N}</math> is a set [[indicator function]]:  <math>1_C(n) = 1</math> iff <math>n \in C</math>, otherwise 0.
*<math>\text{Id}</math> is the identity function with value ''n'':  <math>\text{Id}(n) = n</math>.
*<math>\text{Id}_k</math> is the ''k''th power function:  <math>\text{Id}_k(n)=n^k</math>.

The following relations hold:

* <math>1 * \mu = \varepsilon</math>,&nbsp;the Dirichlet inverse of the constant function <math>1</math> is the [[Möbius function]] (see [[Möbius_function#Proof_of_the_formula_for_the_sum_of_μ_over_divisors|proof]]). Hence:
*<math>g = f * 1</math>  if and only if  <math>f = g * \mu</math>, the [[Möbius inversion formula]].
*<math>\sigma_k = \text{Id}_k * 1</math>, the [[divisor function|kth-power-of-divisors sum function]] ''σ''<sub>''k''</sub>.
*<math>\sigma = \text{Id} * 1</math>, the sum-of-divisors function {{nowrap|1=''σ'' = ''σ''<sub>1</sub>}}.
*<math>\tau = 1 * 1</math> , the number-of-divisors function {{nowrap|1=''τ''(''n'') = ''σ''<sub>0</sub>}}.
*<math>\text{Id}_k = \sigma_k * \mu</math>,&nbsp; by Möbius inversion of the formulas for ''σ''<sub>''k''</sub>, ''σ'', and ''τ''.
*<math>\text{Id} = \sigma * \mu</math>
*<math>1 = \tau * \mu</math>
*<math>\phi * 1 = \text{Id}</math> , proved under [[Euler's totient function#Divisor sum|Euler's totient function]].
*<math>\phi = \text{Id} * \mu</math> , by Möbius inversion.
*<math>\sigma = \phi * \tau</math> &nbsp;, from convolving 1 on both sides of <math>\phi * 1 = \text{Id}</math>.
*<math>\lambda * |\mu| = \varepsilon</math> &nbsp;where ''λ'' is [[Liouville's function]].
*<math>\text{Id} * \phi = P </math>, where <math> P </math> is [[Pillai's arithmetical function]], also known as the gcd-sum function.
*<math>\lambda * 1 = 1_{\text{Sq}}</math>&nbsp;where Sq = {1, 4, 9, ...} is the set of squares.
*<math>\text{Id}_k * (\text{Id}_k \mu) = \varepsilon </math>
*<math>\tau^3 * 1 = (\tau * 1)^2</math>
*<math>J_k * 1 = \text{Id}_k</math>,  [[Jordan's totient function]].
*<math>(\text{Id}_s J_r) * J_s = J_{s + r} </math>
*<math>\Lambda * 1 = \log</math>, where <math>\Lambda</math> is [[von Mangoldt function|von Mangoldt's function]].
*<math>|\mu| \ast 1 = 2^{\omega},</math> where <math>\omega(n)</math> is the [[prime omega function]] counting ''distinct'' prime factors of ''n''.
*<math>\Omega \ast \mu = 1_{\mathcal{P}}</math>, the characteristic function of the prime powers. 
*<math>\omega \ast \mu = 1_{\mathbb{P}}</math> where <math>1_{\mathbb{P}}(n) \mapsto \{0,1\}</math> is the characteristic function of the primes. 

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

:<math>\pi(x) = \sum_{n \leq x} (\omega \ast \mu)(n) = \sum_{d=1}^{x} \omega(d) M\left(\left\lfloor \frac{x}{d} \right\rfloor\right)</math> 

where <math>M(x)</math> is the [[Mertens function]] and <math>\omega</math> 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).<ref>{{cite book|title=Apostol's Introduction to Analytic Number Theory|last1=Schmidt |first1=Maxie}} This identity is a little special something I call "croutons". It follows from several chapters worth of exercises in Apostol's classic book.</ref>
<!-- * ''μ'' ∗ 1 = ''ε'' ∗ (''μ'' ∗ Id<sub>''k''</sub>) ∗ Id<sub>''k''</sub> = ''ε'' (generalized Möbius inversion) -->

==Dirichlet inverse==

===Examples===

Given an arithmetic function <math>f</math> its Dirichlet inverse <math>g = f^{-1} </math> may be calculated recursively: the value of <math>g(n) </math> is in terms of <math>g(m)</math> for <math>m<n</math>.

For <math>n=1</math>:
:<math>(f * g) (1) = f(1) g(1) = \varepsilon(1) = 1 </math>,  so
:<math>g(1) = 1/f(1)</math>. This implies that <math>f</math> does not have a Dirichlet inverse if <math>f(1) = 0</math>.

For <math>n=2</math>:
:<math>(f * g) (2) = f(1) g(2) + f(2) g(1) = \varepsilon(2) = 0</math>,
:<math>g(2) = -(f(2) g(1))/f(1) </math>,

For <math>n=3</math>:
: <math>(f * g) (3) = f(1) g(3) + f(3) g(1) = \varepsilon(3) = 0</math>,
: <math>g(3) = -(f(3) g(1))/f(1) </math>,

For <math>n=4</math>:
: <math>(f * g) (4) = f(1) g(4) + f(2) g(2) + f(4) g(1) = \varepsilon(4) = 0</math>,
: <math>g(4) = -(f(4) g(1) + f(2) g(2))/f(1)  </math>,

and in general for <math>n>1</math>,

:<math>
g(n) \ =\ 
\frac {-1}{f(1)} \mathop{\sum_{d\,\mid \,n}}_{d < n}
f\left(\frac{n}{d}\right) g(d).
</math>

===Properties===

The following properties of the Dirichlet inverse hold:<ref>Again see Apostol Chapter 2 and the exercises at the end of the chapter.</ref>

* The function ''f'' has a Dirichlet inverse if and only if {{nowrap|''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: <math>(f \ast g)^{-1} = f^{-1} \ast g^{-1}</math>. 
* A multiplicative function ''f'' is [[completely multiplicative]] if and only if <math>f^{-1}(n) = \mu(n) f(n)</math>.
* If ''f'' is [[completely multiplicative]] then <math>(f \cdot g)^{-1} = f \cdot g^{-1}</math> whenever <math>g(1) \neq 0</math> and where <math>\cdot</math> denotes pointwise multiplication of functions.

===Other formulas===

{| class="wikitable" border="1"
! Arithmetic function !! Dirichlet inverse:<ref>See Apostol Chapter 2.</ref>
|-
| Constant function with value 1 ||[[Möbius function]] ''μ''
|-
| <math>n^{\alpha}</math> || <math>\mu(n) \,n^\alpha</math>
|-
| [[Liouville's function]] ''λ'' || Absolute value of Möbius function {{abs|''μ''}}
|-
| [[Euler's totient function]] <math>\varphi</math> ||<math>\sum_{d|n} d\, \mu(d)</math>
|-
| The [[sum of divisors|generalized sum-of-divisors function]] <math>\sigma_{\alpha}</math> || <math>\sum_{d|n} d^{\alpha} \mu(d) \mu\left(\frac{n}{d}\right)</math> 
|}

An exact, non-recursive formula for the Dirichlet inverse of any [[arithmetic function]] ''f'' is given in [[Divisor sum identities#The Dirichlet inverse of an arithmetic function|Divisor sum identities]]. A more [[partition theory|partition theoretic]] expression for the Dirichlet inverse of ''f'' is given by 

:<math>f^{-1}(n) = \sum_{k=1}^{\Omega(n)} \left\{ 
     \sum_{{\lambda_1+2\lambda_2+\cdots+k\lambda_k=n} \atop {\lambda_1, \lambda_2, \ldots, \lambda_k | n}} 
     \frac{(\lambda_1+\lambda_2+\cdots+\lambda_k)!}{1! 2! \cdots k!} 
     (-1)^k f(\lambda_1) f(\lambda_2)^2 \cdots f(\lambda_k)^k\right\}.</math>
The following formula provides a compact way of expressing the Dirichlet inverse of an invertible arithmetic function ''f'' :

<math>f^{-1}=\sum_{k=0}^{+\infty}\frac{(f(1)\varepsilon-f)^{*k}}{f(1)^{k+1}}</math>

where the expression <math>(f(1)\varepsilon-f)^{*k}</math> stands for the arithmetic function <math>f(1)\varepsilon-f</math> convoluted with itself ''k'' times. Notice that, for a fixed positive integer <math>n</math>, if <math>k>\Omega(n)</math> then <math>(f(1)\varepsilon-f)^{*k}(n)=0</math> , this is because <math>f(1)\varepsilon(1) - f(1) = 0</math> and every way of expressing ''n'' as a product of ''k'' positive integers must include a 1, so the series on the right hand side converges for every fixed positive integer ''n.'' 

==Dirichlet series==
If ''f'' is an arithmetic function, the [[Dirichlet series]] [[generating function]] is defined by

:<math>
DG(f;s) = \sum_{n=1}^\infty \frac{f(n)}{n^s}
</math>

for those [[complex number|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:

:<math>
DG(f;s) DG(g;s) = DG(f*g;s)\,
</math>

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 concepts==

The restriction of the divisors in the convolution to [[Unitary divisor|unitary]], [[Bi-unitary divisor|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 a special case of the convolution multiplication for the [[incidence algebra]] of a  [[Partially ordered set|poset]], in this case the poset of positive integers ordered by divisibility.

The [[Dirichlet hyperbola method]] computes the summation of a convolution in terms of its functions and their summation functions.

==See also==

* [[Arithmetic function]]
* [[Divisor sum identities]]
* [[Möbius inversion formula]]

==References==

{{Reflist}}
* {{Apostol IANT}}
* {{cite book |
author=Chan, Heng Huat |
title=Analytic Number Theory for Undergraduates |
series=Monographs in Number Theory|
year=2009 |
isbn=978-981-4271-36-3 |
publisher=World Scientific Publishing Company 
}}
* {{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) |author2=Robert C. Vaughan |authorlink2=Robert Charles Vaughan (mathematician)  | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=978-0-521-84903-6 | page=38 | publisher=Cambridge Univ. Press | location=Cambridge }}
* {{Cite news
|first1=Eckford
|last1=Cohen
|title=A class of residue systems (mod r) and related arithmetical functions. I. A generalization of Möbius inversion
|journal=Pacific J. Math.
|volume=9
|number=1
|pages=13–23
|year=1959
|mr=0109806
}}
* {{Cite journal
|first1=Eckford
|last1=Cohen
|title=Arithmetical functions associated with the unitary divisors of an integer
|journal=[[Mathematische Zeitschrift]]
|volume=74
|year=1960
|pages=66–80
|mr=0112861
|doi=10.1007/BF01180473
}}
* {{Cite news
|first1=Eckford
|last1=Cohen
|title=The number of unitary divisors of an integer
|volume=67
|number=9
|pages=879–880
|mr=0122790
|year=1960
|journal=[[American Mathematical Monthly]]
}}
* {{Cite journal
|first1=Graeme L.
|last1=Cohen
|title=On an integers' infinitary divisors
|volume=54
|number=189
|pages=395–411
|mr=0993927
|doi=10.1090/S0025-5718-1990-0993927-5
|journal=Math. Comp.
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* {{Cite journal
|first1=Graeme L.
|last1=Cohen
|title=Arithmetic functions associated with infinitary divisors of an integer
|volume=16
|number=2
|pages=373–383
|doi=10.1155/S0161171293000456
|journal=Int. J. Math. Math. Sci.
|year=1993
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* {{Cite journal
|first1=Pentti
|last1=Haukkanen
|title=Expressions for the Dirichlet inverse of arithmetical functions
|url=https://nntdm.net/volume-06-2000/number-4/118-124/ 
|volume=6
|number=4
|pages=118-124
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|year=2000
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* {{cite journal
|first1=Jozsef
|last1=Sandor
|first2=Antal
|last2=Berge
|title=The Möbius function: generalizations and extensions
|journal=Adv. Stud. Contemp. Math. (Kyungshang)
|volume=6
|number=2
|pages=77–128
|year=2003
|mr=1962765
}}
* {{cite web
 |first1      = Steven
 |last1       = Finch
 |title       = Unitarism and Infinitarism
 |url         = http://www.people.fas.harvard.edu/~sfinch/csolve/try.pdf
 |year        = 2004
 |url-status     = dead
 |archiveurl  = https://web.archive.org/web/20150222094526/http://www.people.fas.harvard.edu/~sfinch/csolve/try.pdf
 |archivedate = 2015-02-22
}}

==External links==
* {{springer|title=Dirichlet convolution|id=p/d130150}}

{{Peter Gustav Lejeune Dirichlet}}

[[Category:Arithmetic functions]]
[[Category:Bilinear maps]]

[[de:Zahlentheoretische Funktion#Faltung]]