View source for Dirichlet convolution

{{TEX|done}}{{MSC|11A25}}

The Dirichlet convolution of two [[arithmetic function]]s $f(n)$ and $g(n)$ is defined as
$$
(f*g)(n) = \sum_{d|n} f(d) g(n/d) \ ,
$$
where the sum is over the positive divisors $d$ of $n$. General background material on the Dirichlet convolution can be found in, e.g., [[#References|[a1]]], [[#References|[a6]]], [[#References|[a8]]].

Sums of the form $\sum_{d|n} f(d) g(n/d)$ played an important role from the very beginning of the theory of arithmetical functions. Many results from early times involved these sums. For example, in 1857 J. Liouville published a long list of arithmetical identities of this type (see [[#References|[a5]]]). It is fruitful to treat the sums $\sum_{d|n} f(d) g(n/d)$ as giving a binary operation on the set of arithmetical functions (cf. also [[Binary relation]]). This aspect was introduced by E.T. Bell [[#References|[a2]]] and M. Cipolla [[#References|[a3]]] in 1915.

The set of arithmetical functions forms a [[commutative ring]] with unity under the usual [[pointwise addition]] and the Dirichlet convolution. An arithmetical function $f$ possesses a Dirichlet inverse if and only if $f(1) \neq 0$. For example, the Dirichlet inverse of the constant function 1 is the [[Möbius function]] $\mu$. The [[Möbius inversion]] formula states that
$$
f(n) = \sum_{d|n} g(d) \Leftrightarrow g(n) = \sum_{d|n} \mu(d) f(n/d) \ .
$$

The relation of the Dirichlet convolution with [[Dirichlet series]] is also important.

There are many analogues and generalizations of the Dirichlet convolution; for example, E. Cohen [[#References|[a4]]] defined the unitary convolution as
$$
(f \otimes g)(n) = \sum_{d \Vert n} f(d) g(n/d) \ ,
$$
where the sum is over the [[unitary divisor]]s of $n$: the positive divisors $d$ of $n$ such that $\mathrm{hcf}(d,n/d) = 1$, see also [[#References|[a10]]]. W. Narkiewicz [[#References|[a7]]] developed a more general convolution:
$$
(f *_A g)(n) = \sum_{d \in A(n)} f(d) g(n/d) \ ,
$$
where, for each $n$, $A(n)$ is a subset of the set of the positive divisors of $n$. See [[#References|[a9]]] for a survey of various binary operations on the set of arithmetical functions.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,   "Introduction to analytic number theory" , Springer  (1976) {{ZBL|0335.10001}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  E.T. Bell,   "An arithmetical theory of certain numerical functions"  ''Univ. Wash. Publ. Math. Phys. Sci.'' , '''I''' :  1  (1915)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Cipolla,   "Sui principi del calculo arithmetico integrale"  ''Atti Accad. Gioenia Cantonia'' , '''5''' :  8  (1915)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Cohen,   "Arithmetical functions associated with the unitary divisors of an integer"  ''Math. Z.'' , '''74'''  (1960)  pp. 66–80</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  L.E. Dickson,   "History of the theory of numbers" , '''I''' , Chelsea, reprint  (1952)</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  P.J. McCarthy,   "Introduction to arithmetical functions" , Springer  (1986)</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  W. Narkiewicz,   "On a class of arithmetical convolutions"  ''Colloq. Math.'' , '''10'''  (1963)  pp. 81–94</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  R. Sivaramakrishnan,   "Classical theory of arithmetic functions" , ''Monographs and Textbooks in Pure and Applied Math.'' , '''126''' , M. Dekker  (1989)</TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top">  M.V. Subbarao,   "On some arithmetic convolutions" , ''The Theory of Arithmetic Functions'' , ''Lecture Notes in Mathematics'' , '''251''' , Springer  (1972)  pp. 247–271</TD></TR>
<TR><TD valign="top">[a10]</TD> <TD valign="top">  R. Vaidyanathaswamy,   "The theory of multiplicative arithmetic functions"  ''Trans. Amer. Math. Soc.'' , '''33'''  (1931)  pp. 579–662</TD></TR>
</table>