Dirichlet Series - Foundations

Dirichlet series provide the crucial bridge between arithmetic functions and complex analysis, transforming discrete number-theoretic questions into problems about analytic functions.

DefinitionDirichlet Series

A Dirichlet series is a series of the form:

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

where $\{a_n\}$ is a sequence of complex numbers and $s \in \mathbb{C}$ is a complex variable. The coefficients $a_n$ typically encode arithmetic information.

ExampleFundamental Examples

Riemann zeta function: $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ (for $\Re(s) > 1$ )
Dirichlet eta function: $\eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s}$
Totient zeta: $\sum_{n=1}^{\infty} \frac{\varphi(n)}{n^s} = \frac{\zeta(s-1)}{\zeta(s)}$ (for $\Re(s) > 2$ )
Divisor series: $\sum_{n=1}^{\infty} \frac{\tau(n)}{n^s} = \zeta(s)^2$ (for $\Re(s) > 1$ )

DefinitionAbscissa of Convergence

The abscissa of absolute convergence $\sigma_a$ is the infimum of real numbers $\sigma$ such that $\sum_{n=1}^{\infty} |a_n|/n^{\sigma} < \infty$ .

The abscissa of convergence $\sigma_c$ is the infimum of real numbers $\sigma$ such that $\sum_{n=1}^{\infty} a_n/n^{\sigma}$ converges.

We always have $\sigma_c \leq \sigma_a \leq \sigma_c + 1$ .

The region of convergence forms a half-plane $\{\Re(s) > \sigma_c\}$ , where the Dirichlet series converges to a holomorphic function. This contrasts with power series, which converge in disks.

Remark

For Dirichlet series with non-negative coefficients, absolute convergence and convergence coincide: $\sigma_c = \sigma_a$ . However, alternating or oscillating coefficients can create a strip of conditional convergence.

ExampleEuler Product Formula

For multiplicative functions $f$ , the Dirichlet series admits an Euler product:

\sum_{n=1}^{\infty} \frac{f(n)}{n^s} = \prod_{p \text{ prime}} \left(1 + \frac{f(p)}{p^s} + \frac{f(p^2)}{p^{2s}} + \cdots \right)

For completely multiplicative $f$ :

\sum_{n=1}^{\infty} \frac{f(n)}{n^s} = \prod_{p} \frac{1}{1 - f(p)p^{-s}}

The Riemann zeta function satisfies $\zeta(s) = \prod_{p} (1 - p^{-s})^{-1}$ , connecting additive summation to multiplicative prime structure.

Dirichlet series transform the algebraic structure of arithmetic functions (Dirichlet convolution) into the multiplicative structure of analytic functions, making them indispensable in modern analytic number theory.