Riemann Zeta Function at Odd Integers

Unsolved Problem

The value of the Riemann $\zeta$ function for odd integers has not been determined.

For certain values it has been calculated to whatever number of places, but an exact expression has never been found.

Examples

Riemann Zeta Function of $3$

Apéry's constant is the value of the infinite sum:

$\map \zeta 3 = \displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$

where $\zeta$ denotes the Riemann zeta function.

Its approximate value is given by:

$\map \zeta 3 \approx 1 \cdotp 20205 \, 69031 \, 59594 \, 28539 \, 97381 \, 61511 \, 44999 \, 07649 \, 86292 \ldots$