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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18
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1answer
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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 ...
6
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2answers
272 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 ...
6
votes
1answer
131 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 $1/2$ seems to hold (as far as I know) for all prime numbers. I was curious if there were any ...
1
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1answer
84 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 ...
1
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0answers
48 views

hypothesis test and execution? did I do this right?

I am trying to figure this out and I'm not sure if I am doing it right. a) did I select the correct type of hypothesis testing??? b) am I using the numbers in the right place (I have SD for 3.5 for ...
2
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0answers
128 views

Mandelbrot set and riemann hypothesis

Has anyone tried to make a connection between the Mandelbrot set and the non-trivial zeros the zeta function? Looking at the Mandelbrot set, it appears that all points are to the left of the line 0.5 ...
4
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2answers
556 views

Andre LeClair, Riemann zeta zero approximation?

This sequence A177885 in the oeis seemingly relates imaginary parts of non-trivial Riemann zeta zeros with the LambertW function. The real and imaginary parts of the Riemann zeta function is the sum ...
14
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1answer
1k 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 ...
2
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0answers
65 views

Validity of a functional formula of the Riemann Zeta function across the whole complex plane?

Could someone confirm me the validity of the following formula: $$\zeta\left(z\right)=2^{z}\pi^{z-1}\sin\left(\frac{\pi z}{2}\right)\left(\frac{e^{-\gamma ...
2
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0answers
144 views

Riemann hypothesis equivalence statement, where is my error?

I need some feedback on the following: According to the page about the von Mangoldt function at the Mathworld page, the Riemann hypothesis is equivalent to the statement: $$\psi = x + ...
5
votes
4answers
4k 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 ...
3
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0answers
79 views

An intuitive interpretation of Montgomery pair corrlation function vs. prime divisibility?

Theorem - If the Riemann hypothesis would be true, and the Montgomery pair correlation conjecture (see linked article page 183-184) true too; let $p \in \Bbb P$ prime, $n \in \Bbb N$ and $$\mathcal ...
5
votes
1answer
166 views

Expanding Riemann Zeta

Consider the Riemann Zeta Function $\zeta(x) = 1 + 2^{-x} + 3^{-x} + 4^{-x}...$ Notice the following identity: $a^{-x} = (e^{ln(a)})^{-x} = e^{-xln(a)}$ Therefore: $\zeta(x) = 1 + 2^{-x} + 3^{-x} ...
4
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0answers
319 views

Riemann $\zeta(s)$ non-trivial zeros on $Re(s)=1/2$ and the “spin” of the primes? [closed]

Can this conjecture be true? Let me intro before getting to the conjcture, but for those who like to go straight, see the bold paragraph at the end. Recently I was reading through the well known ...
4
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2answers
4k views

This is a possible proof of the Riemann Hypothesis [closed]

http://arxiv.org/abs/1305.6845 The above link claims to have solved the Riemann Hypothesis. It's not mine, of course. I just saw this on Tumblr and realized I needed bigger guns. This proof looks like ...
19
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7answers
6k 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 ...
1
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1answer
97 views

Reformulation of riemann zeta

Does this extend to $\mathbb{C}$? $\displaystyle ζ(x) = \int_0^{\infty} \frac{ 1}{\lfloor t\rfloor ^x} dt$, where for $0 \leq t < 1$ we say that $\lfloor t \rfloor = 1$.
-1
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3answers
193 views

Quantum uncertainty can explain the Riemann Hypothesis?

In the recent paper "Riemann Hypothesis as an Uncertainty Relation" (http://arxiv.org/abs/1304.2435) the author claims that the presence of zeros out of the critical line may lead to the violation of ...
8
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3answers
564 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 $\mathbb{Z}$?
-1
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2answers
284 views

Every prime $p_{n}$ is a prime factor of $\frac{1}{2}$ of all the square-free numbers. [closed]

I have removed my bold claims and the naive question so I can link to this post from another. Edit With each prime, we construct square-free numbers that have that prime as the greatest prime ...
26
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1answer
2k 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. Here is my list: The Riemann Hypothesis: A Resource ...
0
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1answer
217 views

Why the Wu-sprung model is not accepted as a solution to Riemann HYpothesis ??

in the Wu-sprung model, given a Hamiltonian in one dimension $$ -y''(x)+f(x)y(x)=E_{n}y(x) \qquad y(0)=0=y(\infty) $$ we can define the function $ f(x) $ implicitly as $$ f^{-1}(x)= 2\sqrt{\pi} ...
3
votes
1answer
99 views

Sums of the negative integer powers of $\zeta$ zeros have an analytical expression…?

The Mathworks page on Riemann's $\zeta$ function says: Let $\rho_k$ denote the $k$th nontrivial zero of $\zeta(s)$, and write the sums of the negative integer powers of such zeros as $$ ...
0
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2answers
132 views

Plot of a Bessel function if possible

i would like to know where i could find a plot of $$ J_{ia}(2\pi i)$$ (1) using Quantum mechanics i have conjectured that if $ a= \frac{x}{2} $ and $ i= \sqrt{-1} $ then $$ J_{it}(2\pi ...
2
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0answers
124 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. Odiuos Number Now consider the sequence of functions below $f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+...$ ...
2
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0answers
146 views

If RH is false , could this be true?

Let $\zeta(s)$ be the Riemann zeta function. Assume RH is false , is it possible that we have in the critical strip $\zeta(a_1+ti) = \zeta(a_2+ti) = \zeta(a_3+ti) = \cdots = \zeta(a_n+ti) = 0$ For ...
4
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3answers
245 views

Riemann Hypothesis: Could there be “simple” ways of getting (partial?) results

Today I did some reading on the Riemann Hypothesis and decided to play around with $\zeta(s)$ a little bit. (In case my question is ridiculous, I'm a student who has no experience dealing with zeta ...
-5
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1answer
204 views

A follow-up problem once the Riemann hypothesis has considered proven to be truth? [closed]

What will happen if a mathmatician have prove that 99.999....% of the solution stay on the critical line and receive the prize but after that another mathmatician find finite numbers of interesting ...
15
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1answer
1k 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 ...
5
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1answer
372 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? ...
4
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1answer
216 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} ...
1
vote
1answer
350 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: ...
15
votes
4answers
667 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 ...
25
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2answers
936 views

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 ...
-2
votes
1answer
280 views

Euler's Choice and Riemann's Oversight?

Euler's Choice: When Euler crafted the zeta function, he knew that $\zeta(1)$ diverged, so he made $\zeta(1)$ undefined. When he crafted the zeta generating function using the Bernoulli numbers, ...
1
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1answer
118 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 ...
2
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1answer
260 views

Factor of the Euler Product at the Roots Of Zeta

The $\zeta$ function maybe written as Euler Product: $$ \zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}=\prod_p e_p(s). $$ Now let's substitute $s$ with $\rho_k$, the $k$th root of $\zeta$, and have a look at ...
4
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3answers
579 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 $ \mathrm{Re}(s) < 1 $? There it is stated that: $$ \sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + ...
5
votes
1answer
470 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 ...
2
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1answer
238 views

Has it been ruled out that the Riemann hypothesis fails for only finite number of zeros?

Has it been ruled out that the Riemann hypothesis fails, but fails only for finite number of zeros?
3
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0answers
308 views

Can the openness of Gilbreath's conjecture be reduced to proof of the Riemann hypothesis?

As an information theoretician, it's a personal hobby of mine to find elegant analogies to open mathematical problems. After all, they have a profound impact on how I conduct my research and how I ...
19
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5answers
3k 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 ...
4
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1answer
294 views

Is there a $p$-adic version of the Riemann hypothesis?

Is there a $p$-adic version of the Riemann hypothesis or this does not make any sense?
3
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3answers
2k views

Riemann Hypothesis and prime number distribution

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 ...
31
votes
4answers
4k 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?
7
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1answer
123 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, ...
8
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3answers
634 views

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 ...
41
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8answers
7k 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?
23
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4answers
1k 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 ...
8
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2answers
657 views

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

(1) The Riemann hypothesis (RH) states that all non-trivial zeros of the zeta function have real part 1/2. (2) The zeta function is intimately connected with the Gamma function via the functional ...