ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
Function
Arithmetic function
Dirichlet character
Dirichlet L-function
Definition D113
Riemann zeta function
Formulation 1
A D4881: Complex function $\zeta : \mathbb{C} \to \mathbb{C}$ is a Riemann zeta function if and only if
(1) $\zeta$ is an D1411: Analytic complex function
(2) \begin{equation} \forall \, z \in \mathbb{C} \left( \Re(z) > 1 \quad \implies \quad \zeta(z) = \lim_{N \to \infty} \sum_{n = 1}^N \frac{1}{n^z} \right) \end{equation}
Formulation 2
A D4881: Complex function $\zeta : \mathbb{C} \to \mathbb{C}$ is a Riemann zeta function if and only if
(1) $\zeta$ is an D1411: Analytic complex function
(2) \begin{equation} \forall \, z \in \mathbb{C} \left( \Re(z) > 1 \quad \implies \quad \zeta(z) = \sum_{n = 1}^{\infty} \frac{1}{n^z} \right) \end{equation}
Formulation 3
A D4881: Complex function $\zeta : \mathbb{C} \to \mathbb{C}$ is a Riemann zeta function if and only if
(1) $\zeta$ is an D1411: Analytic complex function
(2) \begin{equation} \forall \, z \in \mathbb{C} \left( \Re(z) > 1 \quad \implies \quad \zeta(z) = \frac{1}{1^z} + \frac{1}{2^z} + \frac{1}{3^z} + \frac{1}{4^z} + \cdots \right) \end{equation}
Conjectures
Riemann hypothesis