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Riemann Zeta Function 📂Functions

Riemann Zeta Function

Definition

The function defined as ζ:C∖{1}→C\zeta : \mathbb{C} \setminus \left\{ 1 \right\} \to \mathbb{C} is called the Riemann Zeta Function. ζ(s):=∑n∈Nn−s=∏p:prime(1−p−s)−1 \zeta (s) := \sum_{n \in \mathbb{N}} n^{-s} = \prod_{p : \text{prime}} \left( 1- {p^{-s}} \right)^{-1}

  • [0] Ramanujan Sum: If ∑n∈Nxn−1=11−x\displaystyle \sum_{n \in \mathbb{N}} x^{n-1} = {{ 1 } \over { 1-x }} is accepted to hold even at ∣x∣=1|x| = 1, ζ(0)=1+1+1+1+⋯=−12 \zeta (0) = 1 + 1 + 1 + 1 + \cdots = - {{ 1 } \over { 2 }}

  • [1] Ore’s Proof: The reason why ζ(1)\zeta (1) is undefined is as follows. ζ(1)=∑n∈N1n=∞ \zeta (1) = \sum_{n \in \mathbb{N}} {{ 1 } \over { n }} = \infty

  • [2] Euler’s Proof: ζ(2)=∑n∈N1n2=π26 \zeta (2) = \sum_{n \in \mathbb{N}} {{ 1 } \over { n^{2} }} = {{ \pi^{2} } \over { 6 }}

  • [a] Relation with the Gamma Function: If Re⁡(s)>1\operatorname{Re} (s) > 1, then ζ(s)Γ(s)=M[1ex−1](s)=∫0∞xs−1ex−1dx \zeta (s) \Gamma (s) = \mathcal{M} \left[ {{ 1 } \over { e^{x} - 1 }} \right] (s) = \int_{0}^{\infty} {{ x^{s-1} } \over { e^{x} - 1 }} dx

  • [b] Relation with the Dirichlet Eta Function: η(s):=∑n∈N(−1)n−1n−s \eta (s) := \sum_{n \in \mathbb{N}} (-1)^{n-1} n^{-s}

Description

The zeta function converges for complex numbers greater than 11 in real part, that is, for Re⁡(s)>1\operatorname{Re} (s) > 1 within ss and has a relation with the Gamma Function. It has been particularly of interest in number theory and complex analysis and is infamous for being the subject of the Riemann Hypothesis.