ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D113
Riemann zeta function

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) $$\forall \, z \in \mathbb{C} \left( \Re(z) > 1 \quad \implies \quad \zeta(z) = \lim_{N \to \infty} \sum_{n = 1}^N \frac{1}{n^z} \right)$$

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) $$\forall \, z \in \mathbb{C} \left( \Re(z) > 1 \quad \implies \quad \zeta(z) = \sum_{n = 1}^{\infty} \frac{1}{n^z} \right)$$

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) $$\forall \, z \in \mathbb{C} \left( \Re(z) > 1 \quad \implies \quad \zeta(z) = \frac{1}{1^z} + \frac{1}{2^z} + \frac{1}{3^z} + \frac{1}{4^z} + \cdots \right)$$
Conjectures
 ▶ Riemann hypothesis