Riemann Zeta Function - Specific Values

Specific Values

For any positive even number 2n,

where B_2n is a Bernoulli number; for negative integers, one has

for n ≥ 1, so in particular ζ vanishes at the negative even integers because B_m = 0 for all odd m other than 1. No such simple expression is known for odd positive integers.

The values of the zeta function obtained from integral arguments are called zeta constants. The following are the most commonly used values of the Riemann zeta function.

(sequence A059750 in OEIS)

this is employed in calculating of kinetic boundary layer problems of linear kinetic equations.

if we approach from numbers larger 1. Then this is the harmonic series. But its principal value

exists which is the Euler-Mascheroni constant .

(sequence A078434 in OEIS)

this is employed in calculating the critical temperature for a Bose–Einstein condensate in a box with periodic boundary conditions, and for spin wave physics in magnetic systems.

(sequence A013661 in OEIS)

the demonstration of this equality is known as the Basel problem. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime?

(sequence A002117 in OEIS)

this is called Apéry's constant.

(sequence A0013662 in OEIS)

This appears when integrating Planck's law to derive the Stefan–Boltzmann law in physics.