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 Peter Gustav Lejeune Dirichlet.

Definition

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

(f*g)(n)=\sum _{d\,\mid \,n}f(d)g\left({\frac {n}{d}}\right)=\sum _{ab\,=\,n}f(a)g(b)

where the sum extends over all positive divisors d of n, or equivalently over all distinct pairs (a, b) of positive integers whose product is n.

Properties

The set of arithmetic functions forms a commutative ring, the 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 unit function ε defined by ε(n) = 1 if n = 1 and ε(n) = 0 if n > 1. The units (i.e. invertible elements) of this ring are the arithmetic functions f with f(1) ≠ 0.

Specifically, Dirichlet convolution is^[1] associative,

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

distributes over addition

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

,

is commutative,

f*g=g*f

,

and has an identity element,

f*\varepsilon

=

\varepsilon *f=f

.

Furthermore, for each $f$ for which $f(1)\neq 0$ there exists an arithmetic function $f^{-1}$ such that $f*f^{-1}=\varepsilon$ , 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 $f\,(g*h)=(fg)*(fh)$ , where juxtaposition represents pointwise multiplication.^[2] The convolution of two completely multiplicative functions is multiplicative, but not necessarily completely multiplicative.

Examples

In these formulas:

\varepsilon

is the multiplicative identity. (I.e.

\varepsilon (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.

1_C

, where

C\subset \mathbb {Z}

is a set is the indicator function. (I.e.

1_{C}(n)=1

iff

n\in C

.)

Id is the identity function whose value is n. (I.e. Id(n) = n.)

Id_k is the kth power function. (I.e. Id_k(n) = n^k.)

The other functions are defined in the article arithmetical function.

$1*\mu =\varepsilon$ (the Dirichlet inverse of the constant function 1 is the Möbius function.) This implies
g = f ∗ 1 if and only if f = g ∗ μ (the Möbius inversion formula).
λ ∗ |μ| = ε where λ is Liouville's function.
λ ∗ 1 = 1_Sq where Sq = {1, 4, 9, ...} is the set of squares
Id_k ∗ (Id_k μ) = ε
σ_k = Id_k ∗ 1 definition of the divisor function σ_k
σ = Id ∗ 1 definition of the function σ = σ₁
d = 1 ∗ 1 definition of the function d(n) = σ₀
Id_k = σ_k ∗ μ Möbius inversion of the formulas for σ_k, σ, and d.
${\text{Id}}=\sigma *\mu$
$1=d*\mu$
d³ ∗ 1 = (d ∗ 1)²
$\phi *1={\text{Id}}$ This formula is proved in the article Euler's totient function.
J_k ∗ 1 = Id_k The Jordan's totient function.
(Id_sJ_r) ∗ J_s = J_{s + r}
σ = φ ∗ d Proof: convolve 1 to both sides of Id = φ ∗ 1.
Λ ∗ 1 = log where Λ is von Mangoldt function.
$|\mu |\ast 1=2^{\omega },$ where $\omega (n)$ is the prime omega function which counts the number of distinct prime factors of n

Dirichlet inverse

Given an arithmetic function $f$ its Dirichlet inverse $g=f^{-1}$ 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.

For $n=1$ :

(f*g)(1)=f(1)g(1)=\epsilon (1)=1

, so

g(1)=1/f(1)

. This implies that

f

does not have a Dirichlet inverse if

f(1)=0

.

For n = 2

(f ∗ g) (2) = f(1) g(2) + f(2) g(1) = ε(2) = 0,

g(2) = −1/f(1) (f(2) g(1)),

For n = 3

(f ∗ g) (3) = f(1) g(3) + f(3) g(1) = ε(3) = 0,

g(3) = −1/f(1) (f(3) g(1)),

For n = 4

(f ∗ g) (4) = f(1) g(4) + f(2) g(2) + f(4) g(1) = ε(4) = 0,

g(4) = −1/f(1) (f(4) g(1) + f(2) g(2)),

and in general for n > 1,

g(n)={\frac {-1}{f(1)}}\sum _{{\stackrel {d\,\mid \,n}{d<n}}}f\left({\frac {n}{d}}\right)g(d).

Since the only division is by f(1) this shows that f has a Dirichlet inverse if and only if f(1) ≠ 0. An exact, non-recursive formula for the Dirichlet inverse of any arithmetic function f is given in Divisor sum identities.

Here is a useful table of Dirichlet inverses of common arithmetic functions:^[3]

Arithmetic function	Dirichlet inverse
Constant function equal to 1	Möbius function μ
$n^{\alpha}$	$\mu (n)\,n^{\alpha }$
Liouville's function λ	μ\|
Euler's totient function $\varphi$	$\sum _{d\|n}d\mu (d)$
The generalized sum-of-divisors function $\sigma _{\alpha }$	$\sum _{d\|n}d^{\alpha }\mu (d)\mu \left({\frac {n}{d}}\right)$

The following properties of the Dirichlet inverse hold:^[4]

The Dirichlet inverse of a multiplicative function is again multiplicative.
The Dirichlet inverse of a Dirichlet convolution is the convolution of the inverses of each function: $(f\ast g)^{-1}=f^{-1}\ast g^{-1}$ .
A multiplicative function f is completely multiplicative if and only if $f^{-1}(n)=\mu (n)f(n)$ .
If f is completely multiplicative then $(f\cdot g)^{-1}=f\cdot g^{-1}$ whenever $g(1)\neq 1$ and where $\cdot$ denotes pointwise multiplication of functions.

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, etc.).

Dirichlet convolution is the convolution of the incidence algebra for the positive integers ordered by divisibility.

References

↑ Proofs of all these facts are in Chan, ch. 2
↑ A proof is in the article Completely multiplicative function#Proof of distributive property.
↑ See Apostol Chapter 2.
↑ Again see Apostol Chapter 2 and the exercises at the end of the chapter.

Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
Chan Heng Huat (2009). Analytic Number Theory for Undergraduates. Monographs in Number Theory. World Scientific Publishing Company. ISBN 981-4271-36-5.
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. MR 0109806.
Cohen, Eckford (1960). "Arithmetical functions associated with the unitary divisors of an integer". Mathematische Zeitschrift. 74. pp. 66–80. doi:10.1007/BF01180473. MR 0112861.
Cohen, Eckford (1960). "The number of unitary divisors of an integer". American Mathematical Monthly. 67 (9). pp. 879–880. MR 0122790.
Cohen, Graeme L. (1990). "On an integers' infinitary divisors". Math. Comp. 54 (189). pp. 395–411. doi:10.1090/S0025-5718-1990-0993927-5. MR 0993927.
Cohen, Graeme L. (1993). "Arithmetic functions associated with infinitary divisors of an integer". Int. 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. MR 1962765.
Finch, Steven (2004). "Unitarism and Infinitarism" (PDF). Archived from the original (PDF) on 2015-02-22.

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Dirichlet convolution", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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[1] Proofs of all these facts are in Chan, ch. 2

[2] A proof is in the article Completely multiplicative function#Proof of distributive property.

[3] See Apostol Chapter 2.

[4] Again see Apostol Chapter 2 and the exercises at the end of the chapter.