Dirichlet convolution

2020 Mathematics Subject Classification: Primary: 11A25 [MSN][ZBL]

The Dirichlet convolution of two arithmetic functions $f(n)$ 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., [a1], [a6], [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 [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 [a2] and M. Cipolla [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 [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 divisors of $n$: the positive divisors $d$ of $n$ such that $\mathrm{hcf}(d,n/d) = 1$, see also [a10]. W. Narkiewicz [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 [a9] for a survey of various binary operations on the set of arithmetical functions.

References

[a1]	T.M. Apostol, "Introduction to analytic number theory" , Springer (1976) Zbl 0335.10001
[a2]	E.T. Bell, "An arithmetical theory of certain numerical functions" Univ. Wash. Publ. Math. Phys. Sci. , I : 1 (1915)
[a3]	M. Cipolla, "Sui principi del calculo arithmetico integrale" Atti Accad. Gioenia Cantonia , 5 : 8 (1915)
[a4]	E. Cohen, "Arithmetical functions associated with the unitary divisors of an integer" Math. Z. , 74 (1960) pp. 66–80
[a5]	L.E. Dickson, "History of the theory of numbers" , I , Chelsea, reprint (1952)
[a6]	P.J. McCarthy, "Introduction to arithmetical functions" , Springer (1986)
[a7]	W. Narkiewicz, "On a class of arithmetical convolutions" Colloq. Math. , 10 (1963) pp. 81–94
[a8]	R. Sivaramakrishnan, "Classical theory of arithmetic functions" , Monographs and Textbooks in Pure and Applied Math. , 126 , M. Dekker (1989)
[a9]	M.V. Subbarao, "On some arithmetic convolutions" , The Theory of Arithmetic Functions , Lecture Notes in Mathematics , 251 , Springer (1972) pp. 247–271
[a10]	R. Vaidyanathaswamy, "The theory of multiplicative arithmetic functions" Trans. Amer. Math. Soc. , 33 (1931) pp. 579–662