Evaluating Riemann Zeta function( reference request)

Question

I came across the following results. $$\zeta(2)=\sum_{n=1}^\infty\dfrac{1}{n^2}=\dfrac{\pi^2}{6}$$ $$\zeta(4)=\sum_{n=1}^\infty\dfrac{1}{n^4}=\dfrac{\pi^4}{90}$$ $$\zeta(8)=\sum_{n=1}^\infty\dfrac{1}{n^8}=\dfrac{\pi^8}{9450}$$ Are there any good papers/journals or books which rigoursly prove the above results?

Answer 1

Let $B_n$'s are Bernoulli numbers defined by $$\frac{z}{e^z-1}+\dfrac12z=1+\sum_{n=2}^\infty\frac{B_n}{n!}z^n$$ for $n>1$, then for $k>1$ $$\zeta(2k)=(-1)^{k+1}\frac{(2\pi)^{2k}B_{2k}}{2(2k)!}$$ Some of Bernoulli numbers are $$B_0=1,-\frac12,\frac16,0,-\frac{1}{30},0,\frac{1}{42},0,-\frac{1}{30},0,\frac{5}{66},0,\cdots$$ The best Reference I know is

Apostol Tom Mike, Introduction to Analytic Number Theory, Springer-Verlag, New York, (1976)(352s), pp.264-268.

Answer 2

The historical way is showing that $$g(z) = \pi \text{ cotan } \pi z = \frac{1}{z}+ \sum_{n=1}^\infty (\frac{1}{z-n}+\frac{1}{z+n})$$ $$ f(z) = -g'(z) =\frac{\pi^2}{\sin^2 \pi z} = \sum_{n=-\infty}^\infty \frac{1}{(z-n)^2}$$ from which you get $$f^{2k-2}(1/2) = \frac{(2k)!}{2}\sum_{n=-\infty}^\infty \frac{1}{(1/2-n)^{2k}} =(2k)!2^{2k} (1-2^{-2k})\zeta(2k)$$