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value of the Riemann zeta function at
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(Theorem)
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Here we present an application of Parseval's equality to number theory. Let $\zeta(s)$ denote the Riemann zeta function. We will compute the value $$\zeta(2)$$ with the help of Fourier analysis.
Example:
Let $f\colon \Reals \to \Reals$ be the ``identity'' function, defined by $$f(x)=x, \text{ for all }x\in\Reals.$$
The Fourier series of this function has been computed in the entry example of Fourier series.
Thus \begin{eqnarray*} f(x)=\ x&=& a_0^f + \sum_{n=1}^{\infty}(a_n^f\cos(nx)+b_n^f\sin(nx)) \\ &=& \sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{n} \sin(nx), \quad \forall x\in (-\pi,\pi). \end{eqnarray*} Parseval's theorem asserts that:
$$\frac{1}{\pi}\int_{-\pi}^{\pi}f^2(x)dx = 2(a_0^f)^2 + \sum_{k=1}^{\infty}[(a_k^f)^2+(b_k^f)^2].$$
So we apply this to the function $f(x)= x $ $$2(a_0^f)^2 + \sum_{k=1}^{\infty}[(a_k^f)^2+(b_k^f)^2]= \sum_{n=1}^{\infty} \frac{4}{n^2}= 4\sum_{n=1}^{\infty}\frac{1}{n^2}$$ and $$\frac{1}{\pi}\int_{-\pi}^{\pi}f^2(x)dx = \frac{1}{\pi}\int_{-\pi}^{\pi}x^2dx= \frac{2\pi^2}{3}.$$
Hence by Parseval's equality $$4\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{2\pi^2}{3}$$ and hence $$\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}.$$
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Cross-references: Parseval's theorem, example of Fourier series, Fourier series, function, analysis, Riemann zeta function, number theory, Parseval's equality, application
There are 6 references to this entry.
This is version 12 of value of the Riemann zeta function at , born on 2003-09-10, modified 2009-03-14.
Object id is 4719, canonical name is ValueOfTheRiemannZetaFunctionAtS2.
Accessed 9467 times total.
Classification:
| AMS MSC: | 11M99 (Number theory :: Zeta and $L$-functions: analytic theory :: Miscellaneous) | | | 42A16 (Fourier analysis :: Fourier analysis in one variable :: Fourier coefficients, Fourier series of functions with special properties, special Fourier series) |
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Pending Errata and Addenda
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