PlanetMath: value of Dirichlet eta function at $s = 2$

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value of Dirichlet eta function at $s = 2$

(Result)

The value $$\eta(2) = 1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\!-\ldots$$ of the Dirichlet eta function can be found by using the Fourier cosine series expansion of the function $x \mapsto x\!-\!x^2$ on the interval $[0,\,1]$ :

$\displaystyle x\!-\!x^2 \;=\; \frac{1}{6}-\frac{1}{\pi^2}\sum_{n=1}^\infty\frac{\cos{2n\pi x}}{n^2}$ for $\displaystyle \;\; 0 \leqq x \leqq 1$

(1)

Substituting $x := \frac{1}{2}$ to the equation (1) yields $$\frac{1}{4} \;=\; \frac{1}{6}-\frac{1}{\pi^2}\sum_{n=1}^\infty\frac{\cos{n\pi}}{n^2} \;=\; \frac{1}{6}+\frac{1}{\pi^2}\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^2},$$ which we can solve to the form

$\displaystyle \eta(2) \;=\; \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^2} \;=\; \frac{\pi^2}{12}.$

(2)

This result could be obtained very simply by using the functional equation connecting Dirichlet eta function to Riemann zeta function.

Combining the equation (2) with the result concerning the Riemann zeta function at 2 shows that

$\displaystyle 1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\ldots \;=\; \frac{\pi^2}{8}.$

(3)

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Cross-references: Riemann zeta function, functional equation, equation, interval, function, Fourier cosine series, Dirichlet eta function
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This is version 5 of value of Dirichlet eta function at $s = 2$

, born on 2008-09-08, modified 2009-05-18.
Object id is 11010, canonical name is ValueOfDirichletEtaFunctionAtS2.
Accessed 561 times total.

Classification:

AMS MSC:

11M41 (Number theory :: Zeta and $L$-functions: analytic theory :: Other Dirichlet series and zeta functions)

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