Prove-or-disprove-that-S-n-1-1-n-k-k-Z-k-gt-1-is-irrational-

Question Number 3138 by prakash jain last updated on 05/Dec/15

Prove or disprove that

S = \underset{n = 1}{\sum^{\infty}} \frac{1}{n^{k}}, k \in Z^{+}, k > 1 is irrational .

Commented by 123456 last updated on 05/Dec/15

ζ (k)

ζ (2) = \frac{π^{2}}{6} wich is a rational multiple of

π^{2}, so its irational

and in general

ζ (2 n) = q π^{2 n} q \in Q, n \in Z^{+} so its irrational

however nothing is know about

ζ (2 n + 1)

but also

ζ (3) = ζ_{3} is irrational (apery constant)

Commented by Filup last updated on 06/Dec/15

ζ (2 n + 1) = ζ (2 n) (2 n + 1)

Commented by prakash jain last updated on 06/Dec/15

How did you come up with

ζ (2 n + 1) = (2 n + 1) ζ (2 n) ?

From what i know read so far

Funtional Equation

ζ (s) = 2^{s} π^{s - 1} \sin (\frac{π s}{2}) Γ (1 - s) ζ (1 - s)

For s = - 2 n

ζ (- 2 n) = 0

s = + 2 n

\sin (\frac{π s}{2}) Γ (1 - s) \neq 0 so RHS \neq 0

Commented by Filup last updated on 06/Dec/15

sorry, m y m i s t a k e!!!

i was thinking of Γ (2 n + 1) = Γ (2 n) (2 n + 1)

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