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The special-value formula gives $ζ (4) = π^{4} / 90$

Example

$ζ (4) = \frac{π^{4}}{90} .$

Facts & Assumptions

Given: The Bernoulli generating function and the even-integer zeta formula.

[L1]

Bernoulli numbers are defined by $\frac{t}{e^{t} - 1} = \sum_{n \geq 0} \frac{B_{n}}{n!} t^{n}$ (The Bernoulli numbers are defined by the generating series $t / (e^{t} - 1)$ ).

[L2]

For $m \geq 1$ , $ζ (2 m) = (- 1)^{m + 1} \frac{B_{2 m} (2 π)^{2 m}}{2 (2 m)!}$ (The Riemann zeta function has the standard Bernoulli special values at the positive even and nonpositive integers).

Verification

technique · direct

1.1

Expanding $e^{t} - 1 = t + t^{2} / 2 + t^{3} / 6 + t^{4} / 24 + \dots$ and solving for the quotient in [L1] gives $\frac{t}{e^{t} - 1} = 1 - \frac{t}{2} + \frac{t^{2}}{12} - \frac{t^{4}}{720} + \dots,$ so $B_{4} = - 1 / 30$ .

L1givenalgebra

2.1

Substitute $m = 2$ and $B_{4} = - 1 / 30$ into [L2]: $ζ (4) = - \frac{(- 1 / 30) (2 π)^{4}}{2 \cdot 4!} = \frac{π^{4}}{90} .$

step 1.1L2algebra∎

Depends on

Used by

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Sources

K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 11 §3 (standard reference, not scraped)

The special-value formula gives ζ(4)=π4/90ζ(4)=π4/90