zeta2

zeta2 refers to the Riemann zeta function evaluated at the value 2. The Riemann zeta function, denoted by the symbol $\zeta(s)$, is a function of a complex variable $s$ that is defined for $\text{Re}(s) > 1$ by the Dirichlet series:

$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \dots$

This series converges for complex numbers $s$ whose real part is greater than 1. The function can be extended to the entire complex plane, except for a simple pole at $s=1$, through analytic continuation.

The value of zeta2, therefore, is the sum of the reciprocals of the squares of the positive integers:

$\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots$

This sum has a famous closed-form solution, which was first found by Leonhard Euler. The value of zeta2 is $\frac{\pi^2}{6}$. This result is a significant achievement in mathematics, connecting the seemingly disparate fields of number theory and the geometry of circles. It is often referred to as the Basel problem. The exact value $\frac{\pi^2}{6}$ is approximately 1.644934. This value has applications in various areas of physics and mathematics, including statistical mechanics and probability theory.