Dirichlet convolution

In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Johann Peter Gustav Lejeune Dirichlet, a German mathematician.

Definition

If ƒ and g are two arithmetic functions (i.e. functions from the positive integers to the complex numbers), one defines a new arithmetic function ƒ * g, the Dirichlet convolution of ƒ and g, by

$(f*g)(n) = \sum_{d\,\mid \,n} f(d)g(n/d) \,$

where the sum extends over all positive divisors d of n.

Properties

The set of arithmetic functions forms a commutative ring, the Dirichlet ring, under pointwise addition and Dirichlet convolution, with the multiplicative identity given by the function $\epsilon$ defined by $\epsilon$(n) = 1 if n = 1 and $\epsilon$(n) = 0 if n > 1. The units of this ring are the arithmetic functions f with f(1) ≠ 0.

Specifically, Dirichlet convolution is associative,

(f * g) * h = f * (g * h),

distributes over addition

f * (g + h) = f * g + f * h = (g + h) * f,

and is commutative,

f * g = g * f.

Furthermore,

f * $\epsilon$ = $\epsilon$ * f = f,

and for each f for which f(1) ≠ 0 there exists a g such that f * g = $\epsilon$, called the Dirichlet inverse of f.

The Dirichlet convolution of two multiplicative functions 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.

Given a completely multiplicative function f then f (g*h) = (f g)*(f h). The convolution of two completely multiplicative functions need not be completely multiplicative.

Dirichlet inverse

Given an arithmetic function ƒ, an explicit recursive formula for the Dirichlet inverse may be given as follows:

$f^{-1}(1) = \frac {1}{f(1)}$

and for n > 1,

$f^{-1}(n) = \frac {-1}{f(1)}\sum_{d\,\mid \,n,\ d < n} f\left(\frac{n}{d}\right) f^{-1}(d).$

When ƒ(n) = 1 for all n, then the inverse is ƒ ⁻¹(n) = μ(n), the Möbius function. The Möbius inversion formula is the special case of the Dirichlet inversion which is valid for completely multiplicative functions, like f(n)=1.

Dirichlet series

If f is an arithmetic function, one defines its Dirichlet series generating function by

$DG(f;s) = \sum_{n=1}^\infty \frac{f(n)}{n^s}$

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

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,...).

References

• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR0434929 
• Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. 97. Cambridge: Cambridge Univ. Press. p. 38. ISBN 0-521-84903-9. 
• Cohen, Eckford (1959). "A class of residue systems (mod r) and related arithmetical functions. I. A generalization of Möbius inversion". Pacific J. Math. 9 (1): pp. 13—23. MR0109806. 
• Cohen, Eckford (1960). "Arithmetical functions associated with the unitary divisors of an integer". Mathematische Zeitschrift 74: pp. 66—80. doi:10.1007/BF01180473. MR0112861. 
• Cohen, Eckford (1960). "The number of unitary divisors of an integer". American mathematical monthly 67 (9): pp. 879—880. MR0122790. 
• Cohen, Graeme L. (1990). "On an integers' infinitary divisors". Math. Comp. 54 (189): pp. 395—411. doi:10.1090/S0025-5718-1990-0993927-5. MR0993927. 
• Cohen, Graeme L. (1993). "Arithmetic functions associated with infinitary divisors of an integer". Intl. J. Math. Math. Sci. 16 (2): pp. 373—383. doi:10.1155/S0161171293000456. 
• Sandor, Jozsef; Berge, Antal (2003). "The Möbius function: generalizations and extensions". Adv. Stud. Contemp. Math. (Kyungshang) 6 (2): 77–128. MR1962765. 
• Finch, Steven (2004). "Unitarism and Infinitarism". http://algo.inria.fr/csolve/try.pdf.