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## Series with general harmonic number

Calculus (Integrals, Series)
Riemann
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### Series with general harmonic number

Let $\mathcal{H}_n$ denote the $n$ - th harmonic number. It holds that

$$\sum\limits_{n=1}^{\infty}\mathcal{H}_{pn}x^n = -\frac{1}{p}\sum\limits_{k=0}^{p-1} \frac{\ln \varphi_k}{\varphi_k}$$

where $p \in \mathbb{N}$ and $\displaystyle \varphi_k = \varphi_k(x) = 1 - \sqrt[p]{x}\exp\left(\frac{-2\pi ik}{p}\right)$.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$