The connection between the Riemann hypothesis and the harmonic series I have heard most people say that the harmonic series, given by
$\displaystyle\sum_{n=1}^{\infty}{n^{-1}}$, is central or somehow related to a proof of the Riemann hypothesis. I have been trying to understand the sense in which such an assertion is held but to no avail. If anybody could enlighten me about this connection I would be grateful.
The asymptotic of $\sum_{n \le x} \frac1n$ as $x \to \infty$ is easily found with some $x^{-10}$ error term.
The PNT and the Riemann hypothesis (more generally the analytic continuation of $\frac{1}{\zeta(s)},\log \zeta(s),\frac{\zeta'(s)}{\zeta(s)}$) is about the asymptotic of $\prod_{p \le x} \frac{1}{1-p^{-1}} = \sum_{n, \text{Lpf}(n) \le x} \frac1n$ where $\text{Lpf}$ is the largest prime factor.
Well the generalized harmonic series, given by 
$$S_{n,2j}=\sum_{k=1}^n k^{2j}; j\in\mathbb{N}\tag{1},$$ 
is central to a (to be verified) proof of the Riemann hypothesis. See  (Ref.1:a preprint at arXiv).

The method used in Ref.1 is based on the approach of Polya and Hurwitz. It is summarized in a MSE question.

For Generalized Riemann Hypothesis, the Dirichlet character $\chi$ weighted generalized harmonic series appeared. 
$$S_{n,2j}(\chi)=\sum_{k=1}^n \chi(k)k^{2j+b}; b=0,1;j\in\mathbb{N}_0\tag{2}.$$