TeXipedia

Zeta

Represents the uppercase Greek letter used in mathematics, particularly in number theory and complex analysis.

Overview

Primarily associated with the Riemann zeta function, a fundamental concept in analytic number theory and the distribution of prime numbers.

  • Frequently appears in complex analysis and mathematical physics.
  • Used in statistical mechanics and quantum field theory.
  • Common in academic papers discussing special functions and infinite series.
  • Serves as a variable name in mathematical proofs and theoretical physics equations.

Examples

Using the Zeta symbol to represent the Riemann zeta function.

Z(s)=∑n=1∞1ns\Zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} 
\Zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}

Expressing the relationship between the Riemann zeta function and prime numbers.

Z(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 special value of the Riemann zeta function.

Z(2)=π26\Zeta(2) = \frac{\pi^2}{6} 
\Zeta(2) = \frac{\pi^2}{6}