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.

Filter by
Sorted by
Tagged with
76
votes
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 $n^\text{th}$ ...
21
votes
4answers
25k 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 ...
34
votes
4answers
4k views

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 \...
4
votes
1answer
464 views

Testing Zeros Of The Riemann Hypothesis [closed]

I was on Mathworld some time ago when I read this from http://mathworld.wolfram.com/RiemannHypothesis.html: The Riemann hypothesis was computationally tested and found to be true for the first ...
81
votes
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 ...
12
votes
0answers
419 views

What approach provides the largest known verification of the Riemann Hypothesis?

The Riemann Hypothesis is true if and only if Robin's Inequality is true for all n>5040. It has also been shown by Akbary and Friggstad that the smallest counterexample greater than 5040, if it exists,...
55
votes
5answers
37k views

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 ...
22
votes
2answers
624 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 ...
11
votes
2answers
984 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 ...
6
votes
3answers
797 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(\...
68
votes
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?
52
votes
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 ...
18
votes
4answers
950 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}...
24
votes
6answers
7k views

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 ...
52
votes
4answers
12k views

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?
10
votes
3answers
2k views

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 $\mathbb{Z}$?
26
votes
2answers
7k views

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 ...
11
votes
3answers
10k views

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 ...
24
votes
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 ...
11
votes
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 ...
-3
votes
2answers
659 views

Possible progress on the Riemann hypothesis?

Let $\pi(x)$ be the number of primes not exceeding $x$ and $Li(x) = $. Define $Li(x)=\lim_{\epsilon \rightarrow 0^{+}}\Big(\int_{0}^{1-\epsilon} + \int_{1+\epsilon}^{x}\Big) \frac{dt}{\log t} \mathrm{...
7
votes
2answers
879 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 $$|\pi(x)-\rm {li}(x)|\le \frac {1}{8\pi}\sqrt {x}\log (x)\text { for all }x \geq 2657 $$...
1
vote
2answers
401 views

References for Riemann Hypotheis giving the best bound for Prime Number Theorem

Which books cover the proof that Riemann Hypothesis is equivalent to the best error bound for the Prime Number Theorem? My understanding is that Riemann Hypothesis is equivalent to the best bound of ...
5
votes
0answers
168 views

What is the convergence of the explicit formula for $\frac{\zeta'(s)}{\zeta(s)}$?

The explicit formula for $\frac{\zeta'(s)}{\zeta(s)}$ illustrated in (3) below was derived from the relationship illustrated in (1) below using the explicit formula for $\psi(x)$ defined in (2) below. ...
3
votes
0answers
143 views

Questions related to $f(x)$ where the Riemann Xi function $\xi(s)=s\int\limits_0^\infty f(x)\,x^{-s-1}\,dx$

I realize this question is a bit long and contains quite a few formulas, but I believe a considerable amount of background and context are needed to fully understand my questions below. Also, I ...
5
votes
1answer
464 views

The relationship between Golbach's Conjecture and the Riemann Hypothesis

My question pertains to two famous groups of related conjectures: Goldbach's Conjecture (GC); Goldbach's Weak Conjecture (GWC); The Riemann Hypothesis (RH); The Generalized Riemann Hypothesis (GRH). ...
2
votes
0answers
71 views

What is the size of the kth Superabundant Number?

Utilizing the data generated by T.D. Noe for the first $10^6$ terms of A004394, a plot of $k$ against $log_{10}(S(k))$ where S(K) is the $k^{th}$ Superabundant Number shows a strong linear ...
5
votes
1answer
265 views

Riemann Hypothesis numeric verification question?

As I found in wikipedia Riemann Hypothesys has been verified numerically by X. Gourdon (2004) up to 10000000000000 ($10^{13}$) zeroes. I have a few question about how they did it. I tried to read on ...
4
votes
0answers
519 views

A Special Observation on Prime Numbers and $\pi (n)$

$\eth(n)$ is a little algorithm I made, which may appear to be quite complex, so I will start with an example middle of the post. Questions are at the end of the post. Definition Let $...
3
votes
1answer
291 views

Is something similar to Robin's theorem known for possible exceptions to Lagarias' inequality?

Robin's theorem says that if $$\sigma(n)<e^\gamma n\log\log n$$ holds for all $n>5040$, where $\sigma(n)$ is the sum of divisors of $n$, then the Riemann hypothesis is true, but if there are any ...
2
votes
1answer
839 views

How to prove that Riemann zeta function is zero for negative even numbers? [duplicate]

Can anyone please explain to me how to prove that Riemann zeta function is 0 for all negative even numbers. In many references , they have just given the statement without any proof. Any explanation ...
2
votes
1answer
461 views

Elements needed to derive the Riemann-Siegel Z function

On this post I got a comment to motivate the present question: To prove a zero $\zeta(s_0)=0, s_0 \approx 1/2+iy$ is exactly on the critical line, it is enough to prove the real function $Z(t) = \...
22
votes
1answer
3k 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 ...
14
votes
2answers
1k views

A probably wrong proof of the Riemann Hypothesis, but where is the mistake?

There has been a paper doing rounds on Facebook for the past several days, claiming a proof of the Riemann hypothesis. I feel sure that the argument is flawed, but can't see where exactly. It goes as ...
12
votes
0answers
403 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}}} - ...
13
votes
1answer
1k views

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)....
13
votes
8answers
763 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 ...
8
votes
1answer
349 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 ...
5
votes
0answers
295 views

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 ...
6
votes
1answer
949 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 $\frac{1}{2}$. ...
3
votes
2answers
227 views

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 ...
3
votes
3answers
404 views

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: ...
2
votes
1answer
627 views

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: ...
6
votes
0answers
494 views

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 ...
4
votes
1answer
165 views

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 $s>\sigma$, ...
4
votes
0answers
563 views

counterexample to RH; how big would it have to be?

If the Riemann hypothesis is false, then there has to be a first counterexample for $\zeta(z)=0$ in the critical strip with $\Re(z) \ne \frac{1}{2}$. For such a counterexample, how large would $T=|\...
3
votes
0answers
209 views

What is the relationship between GRH and Goldbach Conjecture?

We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, ...
1
vote
1answer
159 views

Distribution of Subsets of Primes

Primes may be divided in to sets: $p=4n\pm1$. Gauss showed, that if $p=4n+1$, it may be written also as $p=a^2+b^2$. From LagrangesFour-SquareTheorem, we know that $g(2)=4$, where 4 may be reduced ...
1
vote
1answer
70 views

A question about an asymptotic formula

I've been told that the asymptotic formula $\pi(x+y)-\pi(x)\sim y/\ln x$ holds for $y\ge x^{1/2+\varepsilon}$ if Riemann's hypothesis is true, but I was unable to find a journal reference for this. ...
0
votes
0answers
47 views

What is asymptotic and error bound for $\sum\limits_{k=1}^K\left(\frac{1}{\rho_k}+\frac{1}{\rho_{-k}}\right)$ as a function of $K$?

This is a follow-on of my previous question What is the convergence of the explicit formula for $\frac{\zeta'(s)}{\zeta(s)}$? My previous question was related to the following two formulas. (1) $\...