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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57 views

Are attempts to solve the Riemann Hypothesis even checked? [on hold]

I am not a mathematician (my speciality is physics) but I’m very interested in math and its open problems. I’ve done some research about the Riemann Hypothesis, and in the process I was amazed to find ...
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25 views

How does the Riemann Hypothesis show the prime spectrum with zeros?

I learned that dependent on the Riemann Hypothesis $$d(x)=-\frac{1}{\pi}\sum_{p^n}\frac{\ln(p)}{p^{\frac{n}{2}}}\cos(x\ln(p^n))$$ has peaks converging at the real points $t$ where $\zeta(\frac{1}{2} + ...
4
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1answer
61 views

Is a strong form of Goldbach conjecture equivalent of Generlized Riemann Hypothesis?

In Andrew Granville's paper: REFINEMENTS OF GOLDBACH’S CONJECTURE, AND THE GENERALIZED RIEMANN HYPOTHESIS He said that: "we show that if a strong form of Goldbach's conjecture is true then every ...
2
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74 views

What are some practical attempts to disprove Riemann Hypothesis?

Most people believe Riemann Hypothesis is true. Since RH has not been proved yet, so it is not completely insane to disprove RH. Among the ways to disprove RH, straightforward ways, such as: try to ...
0
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1answer
63 views

Interpretation help: Showing that Riemann Hypothesis holds “almost surely”

I was perusing this textbook on algorithmic number theory, where I came across this page where they appear to prove that the Riemann Hypothesis holds almost surely. This seems like an odd statement ...
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3answers
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Density of primes containing specific digits

I suspect that primes containing certain digits (e.g. $1$, $3$) are way more common than primes containing other digits e.g. containing $2,4$ since my intuition tells me the latter combination is ...
1
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1answer
60 views

Is the limit of this expression $1$?

Can you help me to determine if the limit of this expression is unity? $$\sum_{n = 2}^{\infty} \frac{ \frac{1}{2n + 1} \cdot \frac{1}{4} \cdot \frac{ 2n}{B_{2n}}}{\sqrt{2}}$$ Where $B_{2n}$ is the ...
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44 views

Showing that a Hermitian matrix can have eigenvalues that correspond to arbitrary numbers does not prove the Hilbert-Polya conjecture, does it?

I read in Wikipedia about the Hilbert-Polya conjecture that: " ...a physical reason that the Riemann hypothesis should be true, and suggested that this would be the case if the imaginary parts ...
2
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1answer
71 views

Mobius function and exp(2 pi i n x)

We know that \begin{equation} \sum_{n \geq 1} \frac{\mu(n)}{n^{s}} = \frac{1}{\zeta(s)}, \end{equation} and so, the left series can be plainly analytically continued to $\text{Re}(s) \leq 1$. ...
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95 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) ...
0
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1answer
138 views

What would the Riemann Hypothesis mean for the Prime Number Theorem?

The Prime Number Theorem states $\pi(n)\sim \dfrac{n}{\ln n}$. Would there be an equally simple expression if Riemann's Hypothesis were proved true? From Chebyshev Function, would $\pi(n)\sim ...
2
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62 views

Proof that the spectrum of prime distribution will give zeros of Riemann Zeta function

All: Many of us have read that the spectrum of prime distribution will give zeros of Riemann Zeta function. For example, Mazur and Stein's book: (http://wstein.org/rh/rh.pdf ) have many nice pictures ...
4
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2answers
106 views

Did Hardy prove that there are countably, or uncountably many zeros on the line Re$(s)=1/2$ of $\zeta(s)$?

It's known that Hardy proved that there are infinitely many zeros of $\zeta(s)$ on the line Re$(s)=\frac{1}{2}$, but did he prove it's countably infinite? Or uncountable?
5
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1answer
110 views

Why Riemann hypothesis and not Riemann's conjecture

I have a stupid question. We say Erdös's conjecture, Goldbach's conjecture, Beal's conjecture... and so on. But we don't say 'Riemann's conjecture.' Instead we use the word 'hypothesis'. Why?
3
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2answers
75 views

Why are the trivial zeros of the Riemann zeta function only negative?

The functional equation of the Riemann zeta function is $$\zeta(s)=2^s\pi^{s-1}\sin(s\pi/2)\Gamma(1-s)\zeta(1-s)$$ clearly $2^s$ and $\pi^{1-s}$ are never equal to zero on the complex plane, and ...
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1answer
138 views

Proof of Riemann Hypothesis

This proof was released this year: http://arxiv.org/abs/1508.00533 Where is the mistake? I just found it and was wondering how obviously wrong it is.
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25 views

Covariance group of the functional equation of an L-function

These last few days, I've been wondering whether one could consider the parameters/variables $\chi$ and $s$ a Dirichlet L-function depends on as coordinates such that the pair of transformations ...
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1answer
65 views

What impact would Riemann's conjecture if it had to be demonstrated as a whole?

I ask myself for many years without ever having answers that suit me. What impact would Riemann's conjecture if it had to be demonstrated as a whole? What would be the impact on our lives? I ...
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134 views

Assuming the Riemann hypothesis, does this integral give the Riemann zeta zeros when increasing Working Precision in Mathematica?

It is probably well known that the Riemann zeta zeros satisfy the following equation: $$\frac{\arg \left(\zeta \left(\rho _n+\frac{1}{1000000000000000}\right)\right)}{\pi }+\frac{\vartheta ...
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Wouldn't the asymptotes of the 2D projection of the inverse of the Riemann Zeta function show the real part of all the non-trivial zeros?

Can somebody provide a visualization of $z=\frac{1}{\zeta(x+iy)}\pm N$ for some large $N$ projected onto the $xz$-plane? I would imagine that if we found any asymptotes converging anywhere other than ...
4
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1answer
113 views

Will Riemann Hypothesis, if true, give us the exact value of number of primes less than a given number?

As we know that RH is the most popular unsolved problem in all of maths. If this hypothesis is true, then will it give us the power to predict the exact number of primes less then a given number? And ...
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1answer
79 views

About one of Riemann's Hypothesis' consequence

In Schoenfeld's (1976) Paper: "Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II", it is shown in Corollary 1. (6.18) that if the Riemann Hypothesis holds, then : $$|\pi (x) ...
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27 views

Selberg's theorem, rotational invariance and circle on the Riemann sphere

If I'm not mistaken, Selberg proved that $\vert\zeta(1/2+it)\vert$ is normally distributed. But the normal distribution is known for its rotational invariance property and as a matter of fact, RH is ...
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What is the origin of this Riemann Hypothesis equivalent involving the Liouville function?

Peter Borwein (in his 2006 book The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, p. 6) provides an equivalence between the Riemann Hypothesis and this conjecture involving ...
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26 views

What are equivalent statements or nature occurrences of Goldbach Conjecture in other math branches? [duplicate]

What are equivalent statements or nature occurrences of Goldbach Conjecture in other math branches ? We all know that Riemann Hypothesis(RH) has many (maybe over one hundred) equivalent statements. ...
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113 views

A mixture with ingredients of two equivalences with Riemann Hypothesis

Let $f(x)=x\cdot(\log x)^x$ for $x\geq 2$, then integrating $\log f(x)=\int_2^x f'(t)/f(t)dt$, it is easy to prove the first statement of following, and directly if we put $x=e^H_n$ and add $H_n$ the ...
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31 views

GRH versus RH - Big picture

What is the relation between generalized Riemann Hypothesis and Riemann Hypothesis? Does proving one have implication the other? Are there results which implies failure of one to failure of other? ...
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146 views

Connection between Riemann hypothesis and distribution of primes. [closed]

Honestly, I have to say that I have hardly any experience in number theory. That's maybe one additional reason why the Riemann hypothesis has such a "mystic" appearance for me. You always hear or read ...
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2answers
206 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 ...
4
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2answers
116 views

Proof of Functional Equation Zeta

$$ \pi^{-s/2}\zeta(s)\Gamma(s/2)=\pi^{-(1-s)/2}\zeta(1-s)\Gamma((1-s)/2) $$ (That's the equation that want to prove) Hello guys, so I'm trying to prove the functional equation of Riemann Zeta, ...
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1answer
75 views

Estimate for partial sums of a series equivalent to the Riemann hypothesis

The sums $$S_N=\sum_{n=1}^N\frac{\mu(n)}{n},$$ where $\mu$ is the Moebius function, are known to tend to 0 as $N\to+\infty$. As far as I remember, there was an estimate on $S_N$ equivalent to the ...
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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 $s>\sigma$, ...
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69 views

How to prove the Riemann hypothesis holds for the first non-trivial zero? [duplicate]

The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta function $\zeta(z)$ lie on the critical line $\Re(z)=1/2$. The MathWorld page on this topic mentions that the hypothesis ...
4
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1answer
59 views

Any correlation to Merten's function?

Here is a plot of partial sums of Liouville Lambda and Moebius Mu: Notice the differences (in green) are tantalizingly close to $-n^{\frac{1}{2}}$. Does this have any correlation to Merten's ...
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51 views

how to prove an approximate to Riemann Xi function having only real zeros

I am searching for approximates to Riemann Xi function. Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via $$\Xi(z)=\xi(1/2+iz)=\xi(s)=(1/2)s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ ...
4
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1answer
104 views

On the series $\sum_{n=1}^{\infty} (H_{n}+\exp(H_{n})\log(H_{n}))/n^{s}$, where $H_{n}$ is the $n$th harmonic number

It is known the following (see [1], here is an open access PDF on his homepage): Theorem (Lagarias, 2002). Let $\sigma(n)$ denote the sum of the positive divisors of $n$. Riemann Hypothesis ...
5
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1answer
145 views

Are there papers or books that explain why Bernhard Riemann believed that his hypothesis is true?

I would like to know what are the mathematical reasons for which Bernhard Riemann believed that his hypothesis is true, and I would like to know if those mathematical reasons were cited in his ...
4
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1answer
372 views

Riemann hypothesis reformulation - again

Yesterday I started to write a paper about the reformulation of the Riemann Hypothesis. My idea was to map the function such that all of the trivial zeros are outside of the unit disk, and the ...
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2answers
107 views

What are the most promising approches for solving Riemann hypothesis? [closed]

I'm not a mathematician but still I'm very interested in Riemann hypothesis. I discovered it with the Numberphile channel. I would like to know what are the current work done of this subject and if ...
3
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2answers
52 views

Bibilography: Riemann's hypothesis and positive semi-definite billinear forms

This is a bibliography request: I remember browsing through a book, some years ago, in a library, in which Riemann's hypothesis was proved over some type of fields (I cannot remember what type), the ...
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33 views

What is general Riemann's Hypothesis? [duplicate]

What makes it so important in analytic number theory?
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81 views

Can $\zeta(s)$ be written in the form $\zeta(s)=\Re(\zeta(s))+i·\Im(\zeta(s)) $ for some subset of $\mathbb{C}$?

Can $\zeta(s)$ be written in the form $\zeta(s)=f(s)+g(s) i $ for some subset of $\mathbb{C}$? I mean, is it possible to develop at least one of the formulas of $\zeta(s)$ so you get something like... ...
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2answers
195 views

Would the Riemann Hypothesis being false affect how frequently primes occur in the number system?

I want to know that if Riemann hypothesis is false (big assumption) would that lead to any effect in how frequently primes occur . Well I got this half cooked information from here: ...
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1answer
607 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 ...
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1answer
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Errors in following paper

In the paper "On the Level Curves of the Xi Function" http://arxiv.org/abs/1002.0352v8, John Breslaw takes a very similar approach to a study of the Riemann hypothesis I did while I was in my first ...
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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 ...
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55 views

Titchmarsh S function

SO it is known that Titchmarsh S function $$ S(T)= \pi^{-1} arg\quad \zeta\bigg(\frac{1}{2}+iT\bigg)$$ under the assumption of *riemann hypothesis * gives $$ S(T)=O(\frac{\log T}{\log \log T})$$ can ...
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292 views

RH would follow from $\displaystyle \frac{p_{n+1}}{p_{n+1}-1}<\frac{\log\log N_{n+1}}{\log\log N_n} $ for all $n>1$; what is my mistake?

Let $N_n=\prod_{k=1}^np_k$ be the primorial of order $n$,$\gamma$ be the Euler-Mascheroni constant and $\varphi$ denote the Euler phi function. Nicolas showed that if the Riemann Hypothesis is true, ...
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1answer
94 views

Some clarifications on analytic continuation of Riemann's Zeta function on $\frac 1 2$

Here's my problem: Riemann's Zeta function converge iff $x>1$ so if I want to have a finite value for $\zeta(\frac 1 2)$ I need to use it's analytic continuation but Riemann's hypothesis states ...
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For Riemann Hypothesis, many people seek physics intuition, why not for Goldbach Conjecture ?

All: As we all know, for Riemann Hypothesis research, many people seek physics intuition, to understand more fundamental reasons why Riemann Hypothesis shall hold. In this direction, we have ...