The connection between the Riemann hypothesis and the harmonic series

Question

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.

Answer 1

• 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.

Answer 2

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. $$S_{n,2j}(\chi)=\sum_{k=1}^n \chi(k)k^{2j+b}; b=0,1;j\in\mathbb{N}_0\tag{2}.$$

Answer 3

This paper proves several equivalents of the Riemann hypothesis in terms of the prime counting function $\pi(x)$ and the harmonic numbers $H_n = \sum_{k = 1}^n \frac{1}{k}$: http://math.colgate.edu/~integers/v31/v31.pdf For example, the Riemann hypothesis is equivalent to $$\left|\mathbf{p}(e^\gamma n) -\frac{1}{n} \sum_{k = 2}^{n-1} \frac{1}{H_k} \right| < \frac{1}{33} \frac{H_n}{\sqrt{n}}+\frac{3}{2n}, \quad \forall n \geq 1,$$ where $\mathbf{p}(x) = \pi(x)/x$.

There's also a famous paper by Lagarias expressing the Riemann hypothesis in terms of the sum of divisors function and the harmonic numbers: https://arxiv.org/abs/math/0008177