where the sum extends over all positive divisorsd 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:
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 functionε 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,^{[1]} Dirichlet convolution is associative,
Furthermore, for each $f$ having $f(1)\neq 0$, there exists an arithmetic function $f^{-1}$ with $f*f^{-1}=\varepsilon$, called the Dirichlet inverse of $f$.
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 $(f+g)(1)=f(1)+g(1)=2\neq 1$), 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$h$, pointwise multiplication by $h$ distributes over Dirichlet convolution: $(f*g)h=(fh)*(gh)$.^{[2]} The convolution of two completely multiplicative functions is multiplicative, but not necessarily completely multiplicative.
$\varepsilon$ is the multiplicative identity: $\varepsilon (1)=1$, otherwise 0 ($\varepsilon (n)=\lfloor {\tfrac {1}{n}}\rfloor$).
$1$ is the constant function with value 1: $1(n)=1$ for all $n$. Keep in mind that $1$ is not the identity. (Some authors denote this as $\zeta$ because the associated Dirichlet series is the Riemann zeta function.)
$1_{C}$ for $C\subset \mathbb {N}$ is a set indicator function: $1_{C}(n)=1$ iff $n\in C$, otherwise 0.
${\text{Id}}$ is the identity function with value n: ${\text{Id}}(n)=n$.
${\text{Id}}_{k}$ is the kth power function: ${\text{Id}}_{k}(n)=n^{k}$.
The following relations hold:
$1*\mu =\varepsilon$, the Dirichlet inverse of the constant function $1$ is the Möbius function (see proof). Hence:
where $M(x)$ is the Mertens function and $\omega$ 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 inverse
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Examples
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Given an arithmetic function $f$ its Dirichlet inverse $g=f^{-1}$ may be calculated recursively: the value of $g(n)$ is in terms of $g(m)$ for $m<n$.
For $n=1$:
$(f*g)(1)=f(1)g(1)=\varepsilon (1)=1$, so
$g(1)=1/f(1)$. This implies that $f$ does not have a Dirichlet inverse if $f(1)=0$.
If f is completely multiplicative then $(f\cdot g)^{-1}=f\cdot g^{-1}$ whenever $g(1)\neq 0$ and where $\cdot$ denotes pointwise multiplication of functions.
where the expression $(f(1)\varepsilon -f)^{*k}$ stands for the arithmetic function $f(1)\varepsilon -f$ convoluted with itself k times. Notice that, for a fixed positive integer $n$, if $k>\Omega (n)$ then $(f(1)\varepsilon -f)^{*k}(n)=0$ , this is because $f(1)\varepsilon (1)-f(1)=0$ 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.
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:
$DG(f;s)DG(g;s)=DG(f*g;s)\,$
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
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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 a special case of the convolution multiplication for the incidence algebra of a 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.
^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.
^Again see Apostol Chapter 2 and the exercises at the end of the chapter.
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Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge: Cambridge Univ. Press. p. 38. ISBN 978-0-521-84903-6.
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Finch, Steven (2004). "Unitarism and Infinitarism" (PDF). Archived from the original (PDF) on 2015-02-22.