Dirichlet series

A Dirichlet series is a series of the form

\sum_{n = 1}^{\infty} \frac{a_{n}}{n^{s}} .

Example: $a_{n} = 1$ gives the Riemann zeta function $ζ (s)$ .

Generalized definition

The classical form is the special case $λ_{n} = \ln n$ of the generalized Dirichlet series (also called Dirichlet series in the sense of Riesz):

\sum_{n = 1}^{\infty} a_{n} e^{- λ_{n} s}, λ_{n} ↗ \infty strictly increasing,

since $n^{- s} = e^{- s \ln n}$ . Any such series is a Dirichlet series; the choice $λ_{n}$ only sets the "spacing" of the exponentials.

Made by Claudio, to be reviewed

Example: the geometric series as a Dirichlet series. With $λ_{n} = n$ , $a_{n} = 1$ and $s = t$ :

\sum_{n = 1}^{\infty} e^{- n t} = \frac{e^{- t}}{1 - e^{- t}} = \frac{1}{e^{t} - 1} .

Converges for all $t > 0$ (arithmetic-progression exponents dominate the $\ln n$ of $ζ$ , improving the abscissa from $σ > 1$ to $σ > 0$ ). Its Mellin transform is the bridge to the classical case:

Γ (s) ζ (s) = \int_{0}^{\infty} \frac{t^{s - 1}}{e^{t} - 1} d t = \sum_{n = 1}^{\infty} \int_{0}^{\infty} t^{s - 1} e^{- n t} d t = Γ (s) \sum_{n = 1}^{\infty} \frac{1}{n^{s}} .

So $\sum e^{- n t}$ and $\sum n^{- s}$ are Mellin-dual faces of the same object. See Fourier, Laplace and Dirichlet relationship for the unified "comb" picture.

Convergence

For a fixed coefficient sequence the convergence region is a half-plane (not a disc, unlike Laurent series):

Abscissa of convergence: $σ_{c} = inf {σ : \sum a_{n} n^{- σ} converges}$ .
Abscissa of absolute convergence: $σ_{a} = inf {σ : \sum | a_{n} | n^{- σ} converges}$ , with $σ_{c} \leq σ_{a} \leq σ_{c} + 1$ .
In every half-plane $Re (s) \geq σ_{c} + ε$ the convergence is uniform, hence the sum is holomorphic there. This is the standard route to the analytic continuation of $ζ$ .
Partial sums obey the Riemann--Stieltjes integral identity

\sum_{n \leq x} a_{n} n^{- s} = \frac{A (x)}{x^{s}} + s \int_{1}^{x} \frac{A (t)}{t^{s + 1}} d t, A (t) = \sum_{n \leq t} a_{n},

which recasts convergence questions in terms of the growth of $A (t)$ .

See absolute convergence and series of functions.

Examples

$a_{n} = 1$ : $ζ (s) = \sum n^{- s}$ (Riemann zeta function).
$a_{n} = (- 1)^{n + 1}$ : $η (s) = \sum (- 1)^{n + 1} n^{- s}$ , the alternating zeta, related to $ζ$ by $η (s) = (1 - 2^{1 - s}) ζ (s)$ .
$a_{n}$ a Dirichlet character: L-functions $L (s, χ)$ , the building blocks of the Dirichlet $L$ -series.
$a_{n} = μ (n)$ (Möbius): $\sum μ (n) n^{- s} = 1 / ζ (s)$ .
$a_{n} = Λ (n) / \ln n$ : $\sum Λ (n) n^{- s} = - ζ^{'} (s) / ζ (s)$ , the logarithmic derivative governing the primes.

Euler product

If $a_{n}$ is multiplicative ( $a_{m n} = a_{m} a_{n}$ for coprime $m, n$ ) and $a_{p^{k}}$ grows mildly, the series factorizes over primes:

\sum_{n = 1}^{\infty} \frac{a_{n}}{n^{s}} = \prod_{p} (1 + \frac{a_{p}}{p^{s}} + \frac{a_{p^{2}}}{p^{2 s}} + \dots) .

For $a_{n} = 1$ this is Euler's product $ζ (s) = \prod_{p} (1 - p^{- s})^{- 1}$ , the analytic bridge to the primes: via Perron's formula it recovers the partial sums $\sum_{n \leq x} a_{n}$ , and the zeros/poles of the Euler product control π(x) and the Chebyshev function. The relation with the logarithmic integral and the location of the zeros of $ζ$ is the content of the prime number theorem and the Riemann hypothesis.

Relation to Fourier series

A Dirichlet series is the Mellin transform of the integer Dirac comb $\sum_{n} a_{n} δ (x - n)$ . Summary: same "basis change" picture as the Laplace/Fourier transforms, but the transform variable enters multiplicatively. See Fourier, Laplace and Dirichlet relationship.