TeXipedia

zeta

Represents the Riemann zeta function, a fundamental mathematical function with deep connections to prime numbers and complex analysis.

Overview

Essential in analytic number theory and complex analysis, serving as a cornerstone for understanding the distribution of prime numbers and numerous mathematical relationships.

  • Plays a central role in the famous Riemann Hypothesis
  • Frequently appears in physics, particularly in quantum field theory and statistical mechanics
  • Used in calculations involving series summations and infinite products
  • Important in studying various mathematical constants and special functions

Examples

Defining the Riemann zeta function for real values greater than 1.

ζ(s)=∑n=1∞1ns,s>1\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \quad s > 1 
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \quad s > 1

Expressing the relationship between the zeta function and prime numbers.

ζ(s)=∏p prime11−p−s\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} 
\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}

Showing a specific value of the zeta function.

ζ(2)=π26≈1.645\zeta(2) = \frac{\pi^2}{6} \approx 1.645 
\zeta(2) = \frac{\pi^2}{6} \approx 1.645