# Questions tagged [riemann-hypothesis]

Questions on the Riemann hypothesis, a conjecture on the behavior of the complex zeros of the Riemann $\zeta$ function. You might want to add the tag [riemann-zeta] to your question as well.

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### The myth of no prime formula?

Terence Tao claims: For instance, we have an exact formula for the $n^\text{th}$ square number – it is $n^2$ – but we do not have a (useful) exact formula for the $n^\text{th}$ ...
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### What is the analytic continuation of the Riemann Zeta Function

I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation. What is this other ...
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### Can someone please explain the Riemann Hypothesis to me… in English?

I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
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### Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. The following are excluded: Books by mathematical ...
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### Where can I find the paper by Guy Robin?

$$\sigma(n) < e^\gamma n \log \log n$$ In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984)....
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### What are some equivalent statements of (strong) Goldbach Conjecture?

What are some equivalent statements of (strong) Goldbach Conjecture ? We all know that Riemann Hypothesis has some interesting equivalent statements. My favorites are involved with Mertens ...
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### (Easy?) consequence of the Riemann Hypothesis

I'm trying to show that the relation $\psi(x)=x+O(\sqrt{x}\log ^2 x)$ (consequence of the Riemann hypothesis) implies $\pi(x)=Li(x)+O(\sqrt{x}\log x)$, where $Li(x)=\int_2^x \frac{dt}{\log t}$. I ...
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### A Thue-Morse Zeta function (Generalized Riemann Zeta function and new GRH)

Consider $t_n$ as the Thue-Morse sequence. Let $m$ be a positive integer and $s$ a complex number, and recall that the Odiuos numbers are the indices of nonzero entries in the Thue-Morse sequence. Now ...
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### How much of the Riemann Hypothesis has been solved?

From Wikipedia, I read ...the Riemann Hypothesis is a conjecture that the Riemann Zeta function has its only zeroes at the negative even integers and complex numbers with real part $\frac{1}{2}$. ...
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### A series whose convergence is equivalent to the Riemann hypothesis

It was claimed here that the convergence of the series$$\sum_{n=2}^\infty \frac{\Lambda(n)-1}{n^{1/2}\log^3 n}\tag1$$(where $\Lambda$ is the Von Mangoldt function) is equivalent to the Riemann ...
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### Riemann explicit formula for $\pi(x)$ and its evaluation

We have for $x>1$: $$\tag{1}\pi^*(x)=R(x)-\sum_{\rho} R(x^{\rho})$$ $\rho$-s are zeros of Riemann zeta function (trivial and nontrivial). I know how to derive $(1)$ and also know how to derive: ...
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### Is classifying one dimensional generalized quasicrystals worthwhile strategy to approach RH?

Works done: After fruitlessly poring over books on zeta functions, it seems Freeman Dyson's sotto voce nudge to classify generalized one-dimensional quasicrystals is a way to go. As he writes: ...
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### Is there an equivalent statement of Riemann Hypothesis in term of Random Matrix or physics theory?

We all know that Riemann Hypothesis has many equivalent statements. After Montgomery’s works on pair-relationship, we now know that ZEROs of Riemann Zeta function has similar properties as ...
### How do you prove that $M(N)=O(N^{1/2+\epsilon})$ from the Riemann Hypothesis?
I understand that if $M(N)=O(N^\sigma)$, then $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}$ and therefore $$\frac{1}{s\zeta(s)} = \int_0^\infty M(x) x^{-(s+1)} dx$$ for $s>\sigma$, ...