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.

For complex numbers $s$ for which $\Re s > 1$, the series

$$\sum_{n = 1}^{\infty} \frac{1}{n^s}$$

converges absolutely and defines an analytic function. The Riemann zeta function is then defined to be the analytic continuation of this function. This continuation has so-called trivial zeros at the negative even integers $-2, -4, -6, ...$ as well as many zeros on the line $\frac{1}{2} + it$. The Riemann hypothesis is a famous conjecture that all the non-trivial zeros of the Riemann zeta function lie on this line.

The Riemann hypothesis has extensive implications in number theory. It is known that the truth of the claim would give precise bounds on the error involved in the prime number theorem, as well as giving strong bounds on the growth of many arithmetic functions (such as the Mertens function). More consequences are listed here.

There has been partial progress towards proving the Riemann hypothesis. Hardy and Littlewood showed that there are infinitely many zeros on the critical line, and that has been improved to show that more than two-fifths of the zeros lie on this line. There is also numerical evidence that the conjecture is true.

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Decidability of the Riemann Hypothesis vs. the Goldbach Conjecture

In the most recent numberphile video, Marcus du Sautoy claims that a proof for the Riemann hypothesis must exist (starts at the 12 minute mark). His reasoning goes as follows: If the hypothesis is undecidable, there is no proof it is false. If we…
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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 $n^\text{th}$ prime number $p_n$! “God may not play dice with the universe, but…
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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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What does proving the Riemann Hypothesis accomplish?

I've recently been reading about the Millenium Prize problems, specifically the Riemann Hypothesis. I'm not near qualified to even grasp the entire problem, but seeing the hypothesis and the other problems I can't help wonder: what is the practical…
Mythio
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Riemann hypothesis: is Bender-Brody-Müller Hamiltonian a new line of attack?

There is a beautiful paper in Physical Review Letters [PRL 118, 130201 (2017), DOI:10.1103/PhysRevLett.118.130201] by Carl Bender, Dorje Brody, and Markus Müller (BBM) on a Hamiltonian approach to the Riemann Hypothesis. The paper is surprisingly…
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Would a proof to the Riemann Hypothesis affect security?

If a solution was found to the Riemann Hypothesis, would it have any effect on the security of things such as RSA protection? Would it make cracking large numbers easier?
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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 cranks (especially books by amateurs who claim to…
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Proof (claimed) for Riemann hypothesis on ArXiv

Has anyone noticed the paper On the zeros of the zeta function and eigenvalue problems by M. R. Pistorius, available on ArXiv? The author claims a proof of RH, and also a growth condition on the zeros. It was posted two weeks ago, and I expected it…
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What is the Todd's function in Atiyah's paper?

In terms of purported proof of Atiyah's Riemann Hypothesis, my question is what is the Todd function that seems to be very important in the proof of Riemann's Hypothesis?
Jose Garcia
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Proving a known zero of the Riemann Zeta has real part exactly 1/2

Much effort has been expended on a famous unsolved problem about the Riemann Zeta function $\zeta(s)$. Not surprisingly, it's called the Riemann hypothesis, which asserts: $$ \zeta(s) = 0 \Rightarrow \operatorname{Re} s = \frac1 2 \text{ or }…
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Colossally abundant numbers and the Riemann hypothesis

[This question has lead me to ask a follow up on MathOverflow.] A recent tweet by John Baez has reminded me of the astonishing fact$^1$ that the Riemann hypothesis (RH) can be disproved by finding a number $n > 5040$ such that $$\frac{\sigma(n)}{n…
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Are there examples that suggest the Riemann Hypothesis might be false?

Are there examples that might suggest the Riemann hypothesis is false? I mean, is there a zeta function $ \zeta (s,X) $ for some mathematical object $X$ with the properties $ \zeta (1-s,X) $ and $ \zeta (s,X)$ are related by a functional…
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Proving the Riemann Hypothesis and Impact on Cryptography

I was talking with a friend last night, and she raised the topic of the Clay Millennium Prize problems. I mentioned that my "favorite" problem is the Riemann Hypothesis; I explained what it posits and mentioned that, if proven, it would have great…
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Would proof of Legendre's conjecture also prove Riemann's hypothesis?

Legendre's conjecture is that there exists a prime number between $n^2$ and $(n+1)^2$. This has been shown to be very likely using computers, but this is merely a heuristic. I have read that if this conjecture is true, the biggest gap between two…
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The main attacks on the Riemann Hypothesis?

Attempts to prove the Riemann Hypothesis So I'm compiling a list of all the attacks and current approaches to Riemann Hypothesis. Can anyone provide me sources (or give their thoughts on possible proofs of it) on promising attacks on Riemann…
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