In mathematics, a Dirichlet series is any series of the form

$\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{a\_n\}\{n^s\},$

where s and a_n, n = 1, 2, 3, ... are complex numbers.

Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Johann Peter Gustav Lejeune Dirichlet.

Examples

---

The most famous of Dirichlet series is

$\backslash zeta(s)=\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{1\}\{n^s\},$

which is the Riemann zeta function. Other Dirichlet series are:

$\backslash frac\{1\}\{\backslash zeta(s)\}=\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{\backslash mu(n)\}\{n^s\}$

where μ(n) is the Möbius function. This and many of the following series may be obtained by applying Möbius inversion and Dirichlet convolution to known series. For example, given a Dirichlet character $\backslash chi(n)$ one has

$\backslash frac\{1\}\{L(\backslash chi,s)\}=\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{\backslash mu(n)\backslash chi(n)\}\{n^s\}$

where $L(\backslash chi,s)$ is a Dirichlet L-function.

Other identities include

$\backslash frac\{\backslash zeta(s-1)\}\{\backslash zeta(s)\}=\backslash sum\_\{n=1\}^\{\backslash infty\}$

\frac{\varphi(n)}{n^s}

where φ(n) is the totient function, and

$\backslash zeta(s)\; \backslash zeta(s-a)=\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{\backslash sigma\_\{a\}(n)\}\{n^s\}$

$\backslash frac\{\backslash zeta(s)\backslash zeta(s-a)\backslash zeta(s-b)\backslash zeta(s-a-b)\}\{\backslash zeta(2s-a-b)\}$

=\sum_{n=1}^{\infty} \frac{\sigma_a(n)\sigma_b(n)}{n^s}

where σ_a(n) is the divisor function. Another divisor function identity is

$\backslash frac\{\backslash zeta^3(s)\}\{\backslash zeta(2s)\}=\backslash sum\_\{n=1\}^\{\backslash infty\}$

\frac{d(n^2)}{n^s}

The logarithm of the zeta function is given by

$\backslash log\; \backslash zeta(s)=\backslash sum\_\{n=2\}^\backslash infty\; \backslash frac\{\backslash Lambda(n)\}\{\backslash log(n)\}\backslash ,\backslash frac\{1\}\{n^s\}$

for $\backslash Re(s)\; >\; 1$. Here, $\backslash Lambda(n)$ is the von Mangoldt function. The logarithmic derivative is then

$\backslash frac\; \{\backslash zeta^\backslash prime(s)\}\{\backslash zeta(s)\}\; =\; -\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash Lambda(n)\}\{n^s\}$

These last two are special cases of a more general relationship for derivatives of Dirichlet series, given below.

Given the Liouville function $\backslash lambda(n)$, one has

$\backslash frac\; \{\backslash zeta(2s)\}\{\backslash zeta(s)\}\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash lambda(n)\}\{n^s\}$

Yet another example involves Ramanujan's sum:

$\backslash frac\{\backslash sigma\_\{1-s\}(m)\}\{\backslash zeta(s)\}=\backslash sum\_\{n=1\}^\backslash infty\backslash frac\{c\_n(m)\}\{n^s\}$

Analytic properties of Dirichlet series

---

Given a sequence {a_n}_{n ∈ N} of complex numbers we try to consider the value of

$f(s)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{a\_n\}\{n^s\}$

as a function of the complex variable s. In order for this to make sense, we need to consider the convergence properties of the above infinite series:

If {a_n}_{n ∈ N} is a bounded sequence of complex numbers, then the corresponding Dirichlet series f converges absolutely on the open half-plane of s such that Re(s) > 1. In general, if $a\_\{n\}\; =\; O(n^\{k\}),$ the series converges absolutely in the half plane $\backslash operatorname\{Re\}(s)\; >\; k\; +\; 1.$

If the set of sums a_n + a_{n + 1} + ... + a_{n + k} is bounded for n and k ≥ 0, then the above infinite series converges on the open half-plane of s such that Re(s) > 0.

In both cases f is an analytic function on the corresponding open half plane.

In general the abscissa of convergence of a Dirichlet series is the intercept on the real axis of the vertical line in the complex line, such that there is convergence to the right of it, and divergence to the left. This is the analogue for Dirichlet series of the radius of convergence for power series. The Dirichlet series case is more complicated, though: absolute convergence and uniform convergence may occur in distinct half-planes.

In many cases, the analytic function associated with a Dirichlet series has an analytic extension to a larger domain.

Derivatives

---

Given

$F(s)\; =\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{f(n)\}\{n^s\}$

for a completely multiplicative function $f(n)$, and assuming the series converges for $\backslash Re(s)\; >\; \backslash sigma\_0$, then one has that

$\backslash frac\; \{F^\backslash prime(s)\}\{F(s)\}\; =\; -\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{f(n)\backslash Lambda(n)\}\{n^s\}$

converges for $\backslash Re(s)\; >\; \backslash sigma\_0$. Here, $\backslash Lambda(n)$ is the von Mangoldt function.

Integral transforms

---

The Mellin transform of a Dirichlet series is given by Perron's formula.

See also

---

• Zeta function regularization

References

---

• Tom Apostol, Introduction to analytic number theory, Springer-Verlag, New York, 1976.

Zeta and L-functions | Mathematical series

Sèrie de Dirichlet | Dirichletreihe | Série de Dirichlet | טור דיריכלה | Dirichlet'n sarja