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| In [[mathematics]], the '''Dirichlet convolution''' is a [[binary operation]] defined for [[arithmetic function]]s; it is important in [[number theory]]. It was developed by [[Peter Gustav Lejeune Dirichlet]], a German mathematician.
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| ==Definition==
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| If ''ƒ'' and ''g'' are two arithmetic functions (i.e. functions from the positive [[integer]]s to the [[complex number]]s), one defines a new arithmetic function ''ƒ'' * ''g'', the ''Dirichlet convolution'' of ''ƒ'' and ''g'', by
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| :<math>
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| \begin{align}
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| (f*g)(n)
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| &= \sum_{d\,\mid \,n} f(d)g\left(\frac{n}{d}\right) \\
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| &= \sum_{ab\,=\,n}f(a)g(b)
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| \end{align}
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| </math>
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| where the sum extends over all positive [[divisor]]s ''d'' of ''n'', or equivalently over all pairs (''a'', ''b'') of positive integers whose product is ''n''.
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| ==Properties==
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| The set of arithmetic functions forms a [[commutative ring]], the '''{{visible anchor|Dirichlet ring}}''', under [[pointwise addition]] (i.e. ''f'' + ''g'' is defined by (''f'' + ''g'')(''n'')= ''f''(''n'') + ''g''(''n'')) and Dirichlet convolution. The multiplicative identity is the function <math>\epsilon</math> defined by <math>\epsilon</math>(''n'') = 1 if ''n'' = 1 and <math>\epsilon</math>(''n'') = 0 if ''n'' > 1. The [[unit (ring theory)|unit]]s (i.e. invertible elements) of this ring are the arithmetic functions ''f'' with ''f''(1) ≠ 0.
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| Specifically, Dirichlet convolution is<ref>Proofs of all these facts are in Chan, ch. 2</ref> [[associativity|associative]],
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| : (''f'' * ''g'') * ''h'' = ''f'' * (''g'' * ''h''),
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| [[distributivity|distributes]] over addition
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| : ''f'' * (''g'' + ''h'') = ''f'' * ''g'' + ''f'' * ''h'' = (''g'' + ''h'') * ''f'',
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| is [[commutativity|commutative]],
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| : ''f'' * ''g'' = ''g'' * ''f'',
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| and has an identity element,
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| : ''f'' * <math>\epsilon</math> = <math>\epsilon</math> * ''f'' = ''f''.
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| Furthermore, for each ''f'' for which ''f''(''1'') ≠ 0 there exists a ''g'' such that ''f'' * ''g'' = <math>\epsilon</math>, called the '''{{visible anchor|Dirichlet inverse}}''' of ''f''.
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| The Dirichlet convolution of two [[multiplicative function]]s is again multiplicative, and every multiplicative function has a Dirichlet inverse that is also multiplicative. The article on multiplicative functions lists several convolution relations among important multiplicative functions.
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| Given a [[completely multiplicative function]] ''f'' then ''f'' (''g''*''h'') = (''f'' ''g'')*(''f'' ''h''), where juxtaposition represents pointwise multiplication.<ref>A proof is in the article [[Completely_multiplicative_function#Proof_of_pseudo-associative_property]].</ref> The convolution of two completely multiplicative functions is ''a fortiori'' multiplicative, but not necessarily completely multiplicative.
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| ==Examples==
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| In these formulas
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| : <math>\epsilon</math> is the multiplicative identity. (I.e. <math>\epsilon</math>(1) = 1, all other values 0.) | |
| : 1 is the constant function whose value is 1 for all ''n''. (I.e. 1(''n'') = 1.) Keep in mind that 1 is not the identity.
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| : 1<sub>''C''</sub>, where <math>\scriptstyle C\subset\mathbb{Z}</math> is a set is the [[indicator function]]. (I.e. 1<sub>''C''</sub>(''n'') = 1 if ''n'' ∈ C, 0 otherwise.)
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| : Id is the identity function whose value is ''n''. (I.e. Id(''n'') = ''n''.)
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| : Id<sub>''k''</sub> is the kth power function. (I.e. Id<sub>''k''</sub>(''n'') = ''n''<sup>''k''</sup>.)
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| : The other functions are defined in the article [[arithmetical function]].
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| * 1 * μ = <math>\epsilon</math> (the Dirichlet inverse of the constant function 1 is the [[Möbius function]].) This implies
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| * ''g'' = ''f'' * 1 if and only if ''f'' = ''g'' * μ (the [[Möbius inversion formula]]).
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| * λ * |μ| = <math>\epsilon</math> where λ is [[Liouville's function]].
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| * λ * 1 = 1<sub>Sq</sub> where Sq = {1, 4, 9, ...} is the set of squares
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| * <math>\sigma</math><sub>''k''</sub> = Id<sub>''k''</sub> * 1 definition of the function σ<sub>''k''</sub>
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| * <math>\sigma</math> = Id * 1 definition of the function σ = σ<sub>1</sub>
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| * ''d'' = 1 * 1 definition of the function ''d''(''n'') = σ<sub>0</sub>
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| * Id<sub>''k''</sub> = <math>\sigma</math><sub>''k''</sub> * <math>\mu</math> Möbius inversion of the formulas for σ<sub>''k''</sub>, σ, and ''d''.
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| * Id = <math>\sigma</math> * <math>\mu</math>
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|
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| * 1 = ''d'' * μ
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| * ''d''<sup> 3</sup> * 1 = (''d'' * 1)<sup>2</sup>
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| * <math>\varphi</math> * 1 = Id This formula is proved in the article [[Euler%27s_totient_function#Divisor_sum|Euler's totient function]].
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| * J<sub>''k''</sub> * 1 = Id<sub>''k''</sub>
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| * (Id<sub>''s''</sub>J<sub>''r''</sub>) * J<sub>''s''</sub> = J<sub>''s'' + ''r''</sub>
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| * <math>\sigma</math> = <math>\varphi</math> * ''d'' Proof: convolve 1 to both sides of Id = <math>\varphi</math> * 1.
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| * Λ * 1 = log where Λ is von Mangoldts' function
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| <!-- * <math>\mu</math> * 1 = <math>\epsilon</math> * (<math>\mu</math> * Id<sub>''k''</sub>) * Id<sub>''k''</sub> = <math>\epsilon</math> (generalized Möbius inversion) -->
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| ==Dirichlet inverse==
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| Given an arithmetic function ''ƒ'' its Dirichlet inverse ''g'' = ''ƒ''<sup>−1</sup> may be calculated recursively (i.e. the value of ''g''(''n'') is in terms of ''g''(''m'') for ''m'' < ''n'') from the definition of Dirichlet inverse.
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| For ''n'' = 1:
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| : (''ƒ'' * ''g'') (1) = ''ƒ''(1) ''g''(1) = <math>\epsilon</math>(1) = 1, so | |
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| : ''g''(1) = 1/''ƒ''(1). This implies that ''ƒ'' does not have a Dirichlet inverse if ''ƒ''(1) = 0.
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| For ''n'' = 2
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| : (''ƒ'' * ''g'') (2) = ''ƒ''(1) ''g''(2) + ''ƒ''(2) ''g''(1) = <math>\epsilon</math>(2) = 0,
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| : ''g''(2) = −1/''ƒ''(1) (''ƒ''(2) ''g''(1)), | |
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| For ''n'' = 3
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| : (''ƒ'' * ''g'') (3) = ''ƒ''(1) ''g''(3) + ''ƒ''(3) ''g''(1) = <math>\epsilon</math>(3) = 0,
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| : ''g''(3) = −1/''ƒ''(1) (''ƒ''(3) ''g''(1)),
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| For ''n'' = 4
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| : (''ƒ'' * ''g'') (4) = ''ƒ''(1) ''g''(4) + ''ƒ''(2) ''g''(2) + ''ƒ''(4) ''g''(1) = <math>\epsilon</math>(4) = 0,
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| : ''g''(4) = −1/''ƒ''(1) (''ƒ''(4) ''g''(1) + ''ƒ''(2) ''g''(2)),
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| and in general for ''n'' > 1,
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| :<math>
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| g(n) =
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| \frac {-1}{f(1)} \sum_\stackrel{d\,\mid \,n} {d < n}
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| f\left(\frac{n}{d}\right) g(d).
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| </math>
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| Since the only division is by ''ƒ''(1) this shows that ''ƒ'' has a Dirichlet inverse if and only if ''ƒ''(1) ≠ 0.
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| Here an useful table of Dirichlet inverses of common arithmetic functions:
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| {| border="1"
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| ! Arithmetic function !! Dirichlet inverse
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| |-
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| | Constant function equal to 1 || [[Möbius function]] <math>\mu</math>
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| |-
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| | <math>n^{\alpha}</math> || <math>\mu(n) \,n^\alpha</math>
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| |-
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| | [[Liouville's function]] <math>\lambda</math> || Absolute value of Möbius function <math>|\mu|</math>
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| |}
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| ==Dirichlet series==
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| If ''f'' is an arithmetic function, one defines its [[Dirichlet series]] [[generating function]] by
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| :<math> | |
| DG(f;s) = \sum_{n=1}^\infty \frac{f(n)}{n^s}
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| </math>
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| 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:
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| :<math>
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| DG(f;s) DG(g;s) = DG(f*g;s)\,
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| </math>
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| for all ''s'' for which both series of the left hand side converge, one of them at least converging
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| 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]].
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| ==Related Concepts==
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| {{expand section|date=December 2013}}
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| 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.).
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| ==References==
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| {{Reflist}}
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| * {{Apostol IANT}}
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| * {{cite book |
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| author=Chan Heng Huat |
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| title=Analytic Number Theory for Undergraduates |
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| series=Monographs in Number Theory|
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| year=2009 |
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| isbn=981-4271-36-5 |
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| publisher=World Scientific Publishing Company
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| }}
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| * {{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) | coauthors=[[Robert Charles Vaughan (mathematician)|Robert C. Vaughan]] | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=0-521-84903-9 | page=38 | publisher=Cambridge Univ. Press | location=Cambridge }}
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| * {{Cite news
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| |first1=Eckford
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| |last1=Cohen
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| |title=A class of residue systems (mod r) and related arithmetical functions. I. A generalization of Möbius inversion
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| |journal=Pacific J. Math.
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| |volume=9
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| |number=1
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| |pages=13—23
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| |year=1959
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| |mr=0109806
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| }}
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| * {{Cite news
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| |first1=Eckford
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| |last1=Cohen
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| |title=Arithmetical functions associated with the unitary divisors of an integer
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| |journal=[[Mathematische Zeitschrift]]
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| |volume=74
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| |year=1960
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| |pages=66—80
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| |mr=0112861
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| |doi=10.1007/BF01180473
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| }}
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| * {{Cite news
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| |first1=Eckford
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| |last1=Cohen
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| |title=The number of unitary divisors of an integer
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| |volume=67
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| |number=9
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| |pages=879—880
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| |mr=0122790
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| |year=1960
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| |journal=[[American mathematical monthly]]
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| }}
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| * {{Cite news
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| |first1=Graeme L.
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| |last1=Cohen
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| |title=On an integers' infinitary divisors
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| |volume=54
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| |number=189
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| |pages=395—411
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| |mr=0993927
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| |doi=10.1090/S0025-5718-1990-0993927-5
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| |journal=Math. Comp.
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| |year=1990
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| }}
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| * {{Cite news
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| |first1=Graeme L.
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| |last1=Cohen
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| |title=Arithmetic functions associated with infinitary divisors of an integer
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| |volume=16
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| |number=2
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| |pages=373—383
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| |doi=10.1155/S0161171293000456
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| |journal=Intl. J. Math. Math. Sci.
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| |year=1993
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| }}
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| * {{cite journal
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| |first1=Jozsef
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| |last1=Sandor
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| |first2=Antal
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| |last2=Berge
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| |title=The Möbius function: generalizations and extensions
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| |journal=Adv. Stud. Contemp. Math. (Kyungshang)
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| |volume=6
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| |number=2
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| |pages=77–128
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| |year=2003
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| |mr=1962765
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| }}
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| * {{cite web
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| |first1=Steven
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| |last1=Finch
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| |title=Unitarism and Infinitarism
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| |url=http://www.people.fas.harvard.edu/~sfinch/csolve/try.pdf
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| |year=2004
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| }}
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| ==External links==
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| * {{springer|title=Dirichlet convolution|id=p/d130150}}
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| [[Category:Number theory]]
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| [[Category:Arithmetic functions]]
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| [[Category:Bilinear operators]]
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| [[Category:Binary operations]]
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| [[de:Zahlentheoretische Funktion#Faltung]]
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