Exercise 6, § 1

Functions of a Real Variable, VII, 327

Let $\psi(x) = \Gamma'(x)/\Gamma(x)$ ; for every integer $q > 1$ and every integer $k$ such that $1 \leq k \leq q-1$ establish the formulae

\sum_{p=1}^{q} \psi\left(\frac{p}{q}\right) \exp\left(\frac{2pk\pi i}{q}\right) = -q \sum_{n=1}^{\infty} \frac{1}{n} \exp\left(\frac{2nk\pi i}{q}\right) = q \log\left(1 - \exp\frac{2k\pi i}{q}\right).

(Use formula (9) of VII, p. 307.)

Solution

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