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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.

  • Note: the Riemann zeta function $\zeta(s)$ at $s=1$ is the (divergent) harmonic series – J. W. Tanner Jun 23 '19 at 23:32
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    http://www.math.lsa.umich.edu/~lagarias/doc/elementaryrh.pdf – Will Jagy Jun 23 '19 at 23:38
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    Well, the Riemann hypothesis is an assertion about the so-called "nontrivial" zeros of the Riemann zeta function $$\zeta(s)=\sum_{n\ge1}\frac1{n^s}.$$ Although the above definitions is for $\text{Re }(s)>1$, one could claim that $$\zeta(1)=\sum_{n\ge1}\frac1n $$ which actually works in the sense that $\zeta(s)$ by definition diverges at $s=1$. Other than the relation between the divergent harmonic series and $\zeta(s)$, there is probably little connection to the Riemann hypothesis. – clathratus Jun 24 '19 at 00:06

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  • 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.

reuns
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  • What do you not understand in my answer ? Depending on the order of summation, $\sum n^{-s}$ goes from "almost trivial" to "as complicated as the RH" – reuns Jun 24 '19 at 05:20
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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

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

mike
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