Riemann zeta function

The Riemann zeta function is defined by

    \[\zeta(s)=\sum\limits_{k=1}^\infty \fr{k^s}=\fr{1^s}+\fr{2^s}+\fr{3^s}+\fr{4^s}+\fr{5^s}+\dots\]

\zeta(1) is the harmonic series.
The values of the zeta function for even integers are known in closed form:

    \begin{align*} \zeta(2)&=\frac{\pi^2}{6} \qquad \text{(Euler)} \\ \zeta(4)&=\frac{\pi^4}{90} \qquad \text{(Euler)} \\ \zeta(6)&=\frac{\pi^6}{945} \qquad \text{(Euler)} \\ \zeta(8)&=\frac{\pi^8}{9450} \\ \zeta(2n)&=\frac{(-1)^{n-1}B_{2n}(2\pi)^{2n}}{2(2n)!} \end{align*}

where B_{2n} are the Bernoulli numbers. From the definition, \zeta(n)\to 1 as n\to\infty; this indicates how B_{2n} grows as n\to\infty.

Deriving sums of other series

    \begin{align*} \intertext{From:} \frac{\zeta(n)}{2^n}&=\fr{2^n}+\fr{4^n}+\fr{6^n}+\dots=\sum\limits_{k=1}^\infty \fr{(2k)^n} \\ \intertext{we have:} (1-\fr{2^n}) \zeta(n)&=1+ \fr{3^n}+\fr{5^n}+\fr{7^n}+\dots=\sum\limits_{k=1}^\infty \fr{(2k-1)^n} \\ \intertext{and:} (1-\fr{2^{n-1}}) \zeta(n)&=1- \fr{2^n}+\fr{3^n}-\fr{4^n}+\dots =\sum\limits_{k=1}^\infty \frac{(-1)^{k-1}}{k^n} \end{align*}


Richard Hamming – Numerical Methods for Scientists and Engineers Ch. 12.4

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