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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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
210 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
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1answer
95 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
127 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
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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. 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
138 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
230 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 ...
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1answer
192 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 ...
5
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1answer
348 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
200 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
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1answer
320 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
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4answers
640 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 ...
22
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2answers
866 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
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1answer
277 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
116 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
257 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
562 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}) + ...
3
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1answer
432 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
225 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?
2
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0answers
280 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 ...
18
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5answers
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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 ...
3
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3answers
1k 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 ...
28
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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?
7
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1answer
119 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
599 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 ...
37
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7answers
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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?
22
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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 ...
8
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2answers
642 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 ...
9
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3answers
974 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 ...