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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80
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4answers
6k views

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 ...
76
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9answers
10k views

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 $n^\text{th}$ ...
69
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7answers
15k views

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?
58
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10answers
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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 ...
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5answers
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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 ...
53
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2answers
10k views

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 ...
51
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2answers
8k views

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 ...
50
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4answers
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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?
43
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1answer
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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?
33
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4answers
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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 \...
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2answers
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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 $ ...
26
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2answers
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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 ...
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6answers
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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 ...
24
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1answer
2k views

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 ...
22
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2answers
613 views

Is there a good (preferably comprehensive) list of which conjectures imply the Riemann Hypothesis?

I wanted to prepare a presentation for the students I tutor on the Clay Millennium problems. This is directed at the Riemann Hypothesis and the Generalized Riemann Hypothesis. The Wikipedia article ...
21
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4answers
24k views

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 ...
21
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1answer
2k views

How not to prove the Riemann hypothesis

I remember reading somewhere that there is a (probably a family of) quick false proof of the Riemann hypothesis that starts by using complex logarithms in a bad way, then does some elementary ...
19
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2answers
673 views

A Neat Identity Involving Zeta Zeroes

While playing around, I encountered the following very curious and cool identity. Consider the exponential integral $\text{Ei}(x)$ and the $n$th nontrivial zero of the Riemann Zeta function $p_n$. ...
18
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4answers
939 views

$\# \{\text{primes}\ 4n+3 \le x\}$ in terms of $\text{Li}(x)$ and roots of Dirichlet $L$-functions

In a paper about Prime Number Races, I found the following (page 14 and 19): This formula, while widely believed to be correct, has not yet been proved. $$ \frac{\int\limits_2^x{\frac{dt}{\ln t}...
17
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0answers
218 views

Related Forms for the Riemann Hypothesis over Finite Fields

There are several formulations and consequences of the Riemann Hypothesis for Curves over Finite Fields. I am interested in the logical implications between those, and in elementary (as possible) ...
16
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1answer
737 views

Consequences of the negation of the Riemann hypothesis

There are many sources documenting the consequences of the Riemann hypothesis, but I can't find one discussing the consequences of its negation, particularly concerning the prime distribution. Can ...
13
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1answer
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Where can I find the paper by Guy Robin?

\begin{equation} \sigma(n) < e^\gamma n \log \log n \end{equation} 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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8answers
737 views

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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0answers
401 views

Are these known telescoping series for $\zeta\left(\frac12\right)$?

There are many known telescoping series for $\zeta(s)$ and I was playing with the following two: $$\displaystyle \zeta(s) = \frac{1}{(s-1)} \left(\sum _{n=1}^{\infty } \left( {\frac {n}{(n+1)^{s}}} - ...
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3answers
2k views

Proving the Riemann Hypothesis without revealing anything other than you proved it

Consider the following assertion from Scott Aaronson's blog: Supposing you do prove the Riemann Hypothesis, it’s possible to convince someone of that fact, without revealing anything other ...
11
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3answers
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How related is the distribution of primes to the Riemann Hypothesis?

I do not grasp all concepts of the Riemann Hypothesis (better yet: as a layman I barely grasp anything). However, I understand that there is a certain link between the Riemann Hypothesis and prime ...
11
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2answers
946 views

Importance of the zero free region of Riemann zeta function

I have heard that for improving the error term in the Prime Number Theorem, we need better and better estimates on the zero free region. I have also heard that the best possible error term comes from ...
11
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2answers
1k views

How to express the Riemann hypothesis in terms of the Gamma function?

The Riemann hypothesis (RH) states that all non-trivial zeros of the zeta function have real part $\frac{1}{2}$. The zeta function is intimately connected with the Gamma function via the functional ...
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3answers
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A cohomological statement equivalent to the Riemann Hypothesis

Is there a possibility for looking for a theory of cohomology and an equivalent cohomological statement for Riemann hypothesis over $\mathbb{Z}$?
10
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3answers
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An inverse for Euler's zeta function product formula

Of course, Euler proved that the Riemann zeta function can be defined as the analytic continuation of a product over all primes. $$\zeta(s) = \prod_{p \in \mathbb{P}}\frac1{1-p^{-s}}$$ It is well ...
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0answers
541 views

Why proving Riemann hypothesis is practically important?

I agree that studying pure mathematics is meaningful by intellectual curiosity itself. However, after AKS algorithm is found, I have a question "Is still Riemann hypothesis practically important ...
9
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4answers
1k views

Usage of Complex Numbers in the Riemann Hypothesis.

I don't have a very good understanding of the Riemann Hypothesis, however that being said, could someone explain to me why complex numbers are used, instead of just using real numbers? Everything I've ...
9
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2answers
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Can you solve this captcha?

I found the following problem in a captcha: (and I was really surprised, I expected just regular blurred or distorted text) What does that mean, and what would the solution be? EDIT: It looks, from ...
9
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1answer
615 views

Packing boxes and proof of Riemann Hypothesis

From Scott Aaronson's blog: There’s a finite (and not unimaginably-large) set of boxes, such that if we knew how to pack those boxes into the trunk of your car, then we’d also know a proof ...
9
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2answers
869 views

Does the Riemann-Hypothesis imply the Twin-Prime-Conjecture?

The Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis) is one of the most important conjectures in number theory. I read that the Riemann hypothesis implies the Goldbach Conjecture ...
9
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0answers
324 views

Can the Riemann hypothesis be relaxed to say that this matrix A consists of square roots?

I realize that asking this question is like presenting to a patent attorney a wheel-less skateboard while asking to patent a hoverboard. Anyways. Lagarias version of the Riemann hypothesis sets a ...
8
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2answers
851 views

Riemann hypothesis and diophantine equation

I read that showing Riemann hypothesis is true was equivalent to showing a particular diophantine equation doesn't have any solution. Is there an explicit example of such a diophantine equation? ...
8
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3answers
401 views

Textbooks for studying Riemann hypothesis

I'm a physics graduate recently learned Riemann hypothesis in a mathematical physics course. ( I knew what the hypothesis is but didn't know mathematical statement) I got interested, and I wanna ...
8
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1answer
344 views

(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 ...
8
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1answer
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Proof by contradiction and ZF set theory

The following is quoted from Sir Michael Atiyah's draft proof of Riemann Hypothesis, Section 5, stating that 'To be explicit, the proof of RH in this paper is by contradiction and this is not accepted ...
7
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1answer
394 views

Is $M(x)=O(x^σ)$ possible with $σ≤1$ even if the Riemann hypothesis is false?

The wiki page on Mertens conjecture and the Connection to the Riemann hypothesis says Using the Mellin inversion theorem we now can express $M$ in terms of 1/ζ as $$ M(x) = \frac{1}{2 \pi i} \...
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2answers
862 views

Riemann Hypothesis and the prime counting function

This article on the prime counting function mentions that the Riemann Hypothesis is equivalent to the statement $$|\pi(x)-\rm {li}(x)|\le \frac {1}{8\pi}\sqrt {x}\log (x)\text { for all }x \geq 2657 $$...
7
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1answer
513 views

Are there any arguments against the Riemann hypothesis?

We all know the well known Riemann hypothesis that the zeroes of the Riemann-zeta function have real part $\frac12$ seems to hold $($as far as I know$)$ for all prime numbers. I was curious if there ...
7
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1answer
158 views

Nonnegativity of the quadratic Dirichlet L-function $L(\tfrac{1}{2},\chi)$ under GRH

I have been looking for a proof of the statement: "Assume the Generalized Riemann Hypothesis. Let $d$ be a fundamental discriminant and $\chi_d$ the associated primitive quadratic character. Then, $$L(...
7
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1answer
175 views

Closed form expression or asymptotic expansion for (periodic) generalized harmonic numbers?

In contrast with the series $\sum_{k=1}^n k$ and $\sum_{k=1}^n1$, there does not (as far as I know) exist a pure closed form expression (or a nice asymptotic expansion other than the Euler-Maclaurin ...
7
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1answer
166 views

parity problems for sieve methods, is it only for Selberg Sieve or for all sieve methods?

It is said that sieve methods have parity problems. Terence Tao gave this "rough" statement of the problem: "Parity problem. If A is a set whose elements are all products of an odd number of primes ...
6
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2answers
2k views

Is this a valid attempt at the Riemann Hypothesis? [closed]

From Marcus Du Sautoy's book “The music of the primes”, there is a method of finding a very long list of N consecutive numbers which are not primes. e.g $101!+2, 101!+3,...,101!+101$ all of which are ...
6
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1answer
924 views

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 $\frac{1}{2}$. ...
6
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3answers
788 views

Does the correctness of Riemann's Hypothesis imply a better bound on $\sum \limits_{p<x}p^{-s}$?

This is follow up question on this: How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $? There it is stated that: $$ \sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + O\left(\...
6
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1answer
479 views

Possible Riemann's Hypothesis proof? [closed]

First of all, I imagine it will not be correct, just because of its simplicity, but I would also want to know why, as I can't find any mistake on it. The "proof" would be based on convining two main ...