Exercise 13, § 2

Functions of a Real Variable, VI, 295

Let f be an integer $\geqslant 1$ , and $\chi$ a function defined on $\mathbf{Z}$ , with complex values, such that $\chi(x + f) = \chi(x)$ for every $x \in \mathbf{Z}$ . If one puts

\hat{\chi}(y) = \frac{1}{f} \sum_{x=0}^{f-1} \chi(x)\,e^{2\pi i y x/f} \quad \text{for } y \in \mathbf{Z},

one has

\chi(x) = \sum_{y=0}^{f-1} \hat{\chi}(y)\, e^{-2\pi i x y/f} \qquad \text{for } x \in \mathbf{Z}.

Deduce from exerc. 12 that, for every integer $m \geqslant 2$ ,

\sum_{n \in \mathbf{Z} \setminus \{0\}} \frac{\chi(n)}{n^m} = -\frac{(2\pi i)^m}{m!} \sum_{y=0}^{f-1} \hat{\chi}(y)\, \mathrm{B}_m\!\left(\frac{y}{f}\right).

For example, one has

1 - \frac{1}{3^3} + \frac{1}{5^3} - \cdots + \frac{(-1)^n}{(2n+1)^3} + \cdots = \frac{\pi^3}{32}.

Solution

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