Apéry’s Constant

Overview

Apéry’s constant is the value of the Riemann zeta function evaluated at the argument 3. It has an approximate value of 1.202.

The Riemann Zeta Function

The Riemann zeta function, denoted by ζ(s), is a function of a complex variable s that generalizes the harmonic series. When s = 3, the series converges to Apéry’s constant.

Significance

Apéry’s constant is a special value of the Riemann zeta function, which is a fundamental function in number theory and has deep connections to the distribution of prime numbers.

Irrationality

Apéry’s constant is an irrational number, meaning it cannot be expressed as a ratio of two integers. This was proven by the French mathematician Roger Apéry in 1978. The exact value of ζ(3) is not known, but it can be approximated using various series expansions and numerical methods.