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 $f,g:\mathbb {N} \to \mathbb {C}$ are two arithmetic functions from the positive integers to the complex numbers, the Dirichlet convolution f ∗ g is a new arithmetic function defined 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.

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:

$\left(\sum _{n\geq 1}{\frac {f(n)}{n^{s}}}\right)\left(\sum _{n\geq 1}{\frac {g(n)}{n^{s}}}\right)\ =\ \left(\sum _{n\geq 1}{\frac {(f*g)(n)}{n^{s}}}\right).$

Properties

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 function ε 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, Dirichlet convolution is 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 $f$ 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 multiplicative function has a Dirichlet inverse which is also multiplicative. 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: $(f*g)h=(fh)*(gh)$. The convolution of two completely multiplicative functions is multiplicative, but not necessarily completely multiplicative.

Examples

In these formulas, we use the following arithmetical functions:

• $\varepsilon$ is the multiplicative identity: $\varepsilon (1)=1$, otherwise 0.
• $1$ 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 $C\subset \mathbb {Z}$ 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. Hence:
• $g=f*1$ if and only if $f=g*\mu$, the Möbius inversion formula
• $\sigma _{k}={\text{Id}}_{k}*1$, the kth-power-of-divisors sum function σ_k
• $\sigma ={\text{Id}}*1$, the sum-of-divisors function σ = σ₁
• $d=1*1$ , the number-of-divisors function d(n) = σ₀
• ${\text{Id}}_{k}=\sigma _{k}*\mu$,  by Möbius inversion of the formulas for σ_k, σ, and d
• ${\text{Id}}=\sigma *\mu$
• $1=d*\mu$
• $\phi *1={\text{Id}}$ , proved under Euler's totient function
• $\phi ={\text{Id}}*\mu$ , by Möbius inversion
• $\sigma =\phi *d$  , from convolving 1 on both sides of $\phi *1={\text{Id}}$
• $\lambda *|\mu |=\epsilon$  where λ is Liouville's function
• $\lambda *1=1_{\text{Sq}}$ where Sq = {1, 4, 9, ...} is the set of squares
• ${\text{Id}}_{k}*({\text{Id}}_{k}\mu )=\epsilon$
• $d^{3}*1=(d*1)^{2}$
• $J_{k}*1={\text{Id}}_{k}$, Jordan's totient function
• $({\text{Id}}_{s}J_{r})*J_{s}=J_{s+r}$
• $\Lambda *1=\log$, where $\Lambda$ is von Mangoldt's function
• $|\mu |\ast 1=2^{\omega },$ where $\omega (n)$ is the prime omega function counting distinct prime factors of n
• $\Omega \ast \mu =\chi _{2{\mathcal {P}}}$ is an indicator function where the set $2{\mathcal {P}}:=\{n\geq 1:n=2^{k}\vee n\in \mathbb {P} \}$ is the collection of positive primes and integral powers of two.
• $\omega \ast \mu =\chi _{\mathbb {P} }$ where $\chi _{\mathbb {P} }(n)\mapsto \{0,1\}$ is the characteristic function of the primes.

This last identity shows that the prime counting function is given by the summatory function $\pi (x)=\sum _{n\leq x}(\omega \ast \mu )(n)=\sum _{d=1}^{x}\omega (d)M\left(\left\lfloor {\frac {x}{d}}\right\rfloor \right)$ 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 concolutions given on the divisor sum identities page (a standard trick for these sums).

Dirichlet inverse

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 $n=1$:

$(f*g)(1)=f(1)g(1)=\epsilon (1)=1$, so

$g(1)=1/f(1)$. This implies that $f$ does not have a Dirichlet inverse if $f(1)=0$.

For $n=2$:

$(f*g)(2)=f(1)g(2)+f(2)g(1)=\epsilon (2)=0$,

$g(2)=-1/f(1)(f(2)g(1))$,

For $n=3$:

$(f*g)(3)=f(1)g(3)+f(3)g(1)=\epsilon (3)=0$,

$g(3)=-1/f(1)(f(3)g(1))$,

For $n=4$:

$(f*g)(4)=f(1)g(4)+f(2)g(2)+f(4)g(1)=\epsilon (4)=0$,

$g(4)=-1/f(1)(f(4)g(1)+f(2)g(2))$,

and in general for $n>1$,

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

Since the only division is by f(1), this shows 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.

Arithmetic function	Dirichlet inverse:
Constant function with value 1	Möbius function μ
$n^{\alpha }$	$\mu (n)\,n^{\alpha }$
Liouville's function λ	Absolute value of Möbius function |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:

• 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 $g(1)\neq 0$ 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.

See also

• Arithmetic function
• Divisor sum identities
• Möbius inversion formula

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, 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. Vol. 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. Vol. 9, no. 1. pp. 13–23. MR 0109806.
• Cohen, Eckford (1960). "Arithmetical functions associated with the unitary divisors of an integer". Mathematische Zeitschrift. Vol. 74. pp. 66–80. doi:10.1007/BF01180473. MR 0112861.
• Cohen, Eckford (1960). "The number of unitary divisors of an integer". American Mathematical Monthly. Vol. 67, no. 9. pp. 879–880. MR 0122790.
• Cohen, Graeme L. (1990). "On an integers' infinitary divisors". Math. Comp. Vol. 54, no. 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. Vol. 16, no. 2. pp. 373–383. doi:10.1155/S0161171293000456.{{cite news}}: CS1 maint: unflagged free DOI (link)
• 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. {{cite web}}: Unknown parameter |deadurl= ignored (|url-status= suggested) (help)

External links

• "Dirichlet convolution", Encyclopedia of Mathematics, EMS Press, 2001 [1994]