Welcome to mathimatikoi.org;a forum of university mathematics. Enjoy your stay here.

Series with general harmonic number

Calculus (Integrals, Series)
Post Reply
User avatar
Riemann
Articles: 0
Posts: 168
Joined: Sat Nov 14, 2015 6:32 am
Location: Melbourne, Australia

Series with general harmonic number

#1

Post by Riemann » Sun Aug 12, 2018 8:27 pm

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