Dirichlet series

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

$\sum_{n=1}^{\infty} \frac{a_n}{n^s},$

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.

1 Examples
2 Analytic properties of Dirichlet series: the abscissa of convergence
3 Derivatives
4 Products
5 Integral transforms
6 See also
7 References

[edit] Examples

The most famous of Dirichlet series is

$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s},$

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

$\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}$

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 $χ(n)$ one has

$\frac{1}{L(\chi,s)}=\sum_{n=1}^{\infty} \frac{\mu(n)\chi(n)}{n^s}$

where $L (χ, s)$ is a Dirichlet L-function.

Other identities include

$\frac{\zeta(s-1)}{\zeta(s)}=\sum_{n=1}^{\infty} \frac{\varphi(n)}{n^s}$

where φ(n) is the totient function, and

$\zeta(s) \zeta(s-a)=\sum_{n=1}^{\infty} \frac{\sigma_{a}(n)}{n^s}$

$\frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)} =\sum_{n=1}^{\infty} \frac{\sigma_a(n)\sigma_b(n)}{n^s}$

where σ_a(n) is the divisor function. Other identities involving the divisor function d=σ₀ are

$\frac{\zeta^3(s)}{\zeta(2s)}=\sum_{n=1}^{\infty}\frac{d(n^2)}{n^s}$

$\frac{\zeta^4(s)}{\zeta(2s)}=\sum_{n=1}^{\infty}\frac{d(n)^2}{n^s}$

The logarithm of the zeta function is given by

$\log \zeta(s)=\sum_{n=2}^\infty \frac{\Lambda(n)}{\log(n)}\,\frac{1}{n^s}$

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

$\frac {\zeta^\prime(s)}{\zeta(s)} = -\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}$

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

Given the Liouville function $λ(n)$ , one has

$\frac {\zeta(2s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s}$

Yet another example involves Ramanujan's sum:

$\frac{\sigma_{1-s}(m)}{\zeta(s)}=\sum_{n=1}^\infty\frac{c_n(m)}{n^s}$

[edit] Analytic properties of Dirichlet series: the abscissa of convergence

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

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

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 $\operatorname{Re}(s) > k + 1.$

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 plane, 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.

[edit] Derivatives

Given

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

for a completely multiplicative function $f (n)$ , and assuming the series converges for $\Re(s) > \sigma_0$ , then one has that

$\frac {F^\prime(s)}{F(s)} = - \sum_{n=1}^\infty \frac{f(n)\Lambda(n)}{n^s}$

converges for $\Re(s) > \sigma_0$ . Here, $Λ(n)$ is the von Mangoldt function.

[edit] Products

Let be $F(s)= \sum_{n=1}^{\infty} f(n)n^{-s}$ and

$G(s)= \sum_{n=1}^{\infty} g(n)n^{-s}$

If both F(s) and G(s) are absolutely convergent for s> a and s > b then we have that:

$\frac{1}{2T}\int_{-T}^{T}dtF(a+it)G(b-it)dt= \sum_{n=1}^{\infty} f(n)g(n)n^{-a-b}$ as $T \sim \infty$

for a=b and f(n)=g(n) we have:

$\frac{1}{2T}\int_{-T}^{T}dt|F(a+it)|^{2} dt= \sum_{n=1}^{\infty} [f(n)]^{2}n^{-2a}$ as $T \sim \infty$

[edit] Integral transforms

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

[edit] See also

Zeta function regularization

[edit] References

↑ Tom Apostol, Introduction to analytic number theory, Springer-Verlag, New York, 1976.
G. H. Hardy, and Marcel Riesz, The general theory of Dirichlet's series, Cambridge Tracts in Mathematics, No. 18 (Cambridge University Press, 1915).
The general theory of Dirichlet's series by G. H. Hardy. Cornell University Library Historical Math Monographs. {Reprinted by} Cornell University Library Digital Collections