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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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213 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) ...
12
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398 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}}} - ...
10
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536 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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323 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 ...
6
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158 views

Does $N_0(T) =N(T)$ where $N$ is the exact Riemann $\zeta$ zero counting function and $N_0$ is the approximate zero counting function imply the RH?

From A theory for the zeros of Riemann ζ and other L-functions The main new result is that the zeros on the critical line are in one-to- one correspondence with the zeros of the cosine ...
6
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467 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 ...
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133 views

Can the Riemann Explicit Formula be used to find prime numbers?

It is well known that there is a strong link between the Riemann Hypothesis and the distribution of primes. The prime number theorem gives the number of primes less than or equal to a given $N$ as: ...
5
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289 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 ...
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43 views

Is there any numerical data or theorems on the minimum distance between consecutive zeros of the zeta function?

Title basically. I'm looking for any results, papers, or data pertaining to the distribution of zeros of the zeta function on the critical line. Some numerical data which has a max height $T$ and $\...
4
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247 views

Is there a hidden connection between RH and the golden ratio?

I realized today that, considering the circle $ \Gamma_{\Delta} $ on the Riemann sphere whose image through the stereographic projection is the critical line $ \Delta $, the affixes of the images of ...
4
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166 views

Question on Integral Transform Related to Riemann Zeta Function $\zeta(s)$

The question below assumes the following definitions. $\quad\zeta(s)$ - Riemann zeta function $\quad\psi(x)$ - second Chebyshev function $\quad J(x)$ - Riemann prime-power counting function The ...
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513 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=|\...
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499 views

Symmetry and the zeta function

It is an incontestable fact that symmetry is one of, if not the, most powerful tools a mathematician can have at his or her disposal. What I want to know, is if there are any good reasons as to why it ...
3
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129 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 ...
3
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140 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
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77 views

Are there any theorem relating transcendency/irrationality of Euler's constant and the Riemann hypothesis?

Several formulations of the Riemann hypothesis can be given in terms of Euler’s constant. For example, Theorem (Nicolas 1981) The Riemann hypothesis holds if and only if all the primorial numbers $...
3
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143 views

A possible numerical argument for the Riemann hypothesis

According to an answer on this MO post, showing that $$\int_{0}^{\infty}\frac{(1-12t^2)}{(1+4t^2)^3}\int_{1/2}^{\infty}\log|\zeta(\sigma+it)|~d\sigma ~dt=\frac{\pi(3-\gamma)}{32}$$ $($$\gamma$ is the ...
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356 views

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

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 that ...
3
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73 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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200 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, ...
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196 views

Consequence of the Riemann Hypothesis

So I watched this video: http://m.youtube.com/watch?v=rGo2hsoJSbo And it included the fact that a consequence of RH is that there will always be a prime number between consecutive cubic numbers. I ...
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306 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 + \mathcal{O}(\...
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117 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 ...
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309 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?
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39 views

Some inequalities similar to Lagarias inequality

I have found some inequalities which seem to be equivalent to Lagarias inequality. It would be very nice if someone takes the time to proofread this in detail and give constructive feedback. Thanks ...
2
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91 views

The Meaning of R's non-Trivial Zeros

(I have read through the various similar questions on SE listed by the system but not found an answer that helps). Is there an intuitive explanation for why the Riemann zeta Function (rather than ...
2
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0answers
104 views

What is the mass center of the Riemann Zeta Function across the critical line?

I just came with the idea: what is the center of mass of the Riemann Zeta Function across the critical line? I mean: when you plot the parametric graph across the critical line, you get the famous ...
2
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118 views

Questions on Riemann's Prime-Power Counting Function $\Pi(x)$ and a Related Staircase Function

This question is related to the prime-counting function $\Pi(x)$ and staircase function $Q(x)$ defined in (1) and (2) below respectively. (1) $\quad\Pi(x)=\sum\limits_{n=2}^\infty\frac{\Lambda(n)}{\...
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0answers
222 views

Riemann-Zeta Zeros and Quasicrystals

I came across quasicrystals in the Wikipedia page for the Riemann Hypothesis and then followed the references. On page 215 of Birds and Frogs Dyson makes the claim If the Riemann hypothesis is true,...
2
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98 views

How is the Riemann zeta function $\zeta(s)$ determined by its values in a small open disc?

This question pertains to the following quote from chapter 7 section 9 of "Summing it Up: From One Plus One to Modern Number Theory" by Avner Ash and Robert Gross: "What makes the Riemann ...
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117 views

Questions on Evaluation of Integral for Recovering Second Chebyshev Function $\psi[y]$

The context of this question is the relationship between evaluation of the partial integral for $\psi(x)$ defined in (1) below and partial evaluation of von Mandgoldt's explicit formula for $\psi(x)$ ...
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111 views

Questions on Relationships for Recovering $J[y]$ and $\psi(y)$ from Functions of $\zeta[s]$

I've been exploring recovering Riemann's prime-power counting function $J[y]$ and the second Chebyshev function $\psi(y)$ from functions of $\zeta[s]$ via relationships such as the following. $$J(y)=\...
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127 views

Has anyone seen this divisor function identity?

I worked this out while working on the Lagarias/Robin/Nicolas inequalities: $$\sigma(n) = \frac{n^2}{\phi(n)}\cdot\prod_{p|n}{\left(1-\frac{1}{p^{k+1}}\right)}$$ where $k$ is the largest exponent of ...
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223 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 ...
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303 views

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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111 views

Similarity of two limits related to the sum of divisors $\sigma(n)$ and the harmonic numbers $H_n$

Given that the sum of divisors has the form: $$\large \sigma(n) = \sum _{k=1}^n \lim_{s\to 0} \, \left(\frac{(s+1) (-1)^{\frac{2 n}{k}}+s-1}{k \cdot s \cdot 2}\right)^{-1}$$ $$1, 3, 4, 7, 6, 12, 8, ...
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82 views

Prime bounds under RH

Continuing from here, since $$ \sum_{k=1}^{\lfloor\log(n)\rfloor}\dfrac{\pi(n^{1/k})}{k}=\operatorname{li}(n)-\sum_{k=1}^{\infty}2\ \Re\left(\operatorname{Ei}\left(\rho_k\log\left(n\right)\right)\...
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69 views

Periodicity in Riemann zeros.

Has someone studied if the non-trivial zeroes of the Riemann zeta function has some "periodicity" or "quasiperiodicity"? And what about generalized zeta functions and/or L-functions?
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112 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 \left(1-z\right)}}{1-z}\prod_{n=1}^{\infty}\...
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252 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 $...
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593 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 ...
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70 views

RH & explicit formula for the number of primes $ \le x$

Does the RH have to be true in order for Riemann's explicit formula for the number of primes <= x to hold? The formula is (copied from wikipedia:https://en.wikipedia.org/wiki/Explicit_formulae_(L-...
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81 views

A group theoretic interpretation of Lagarias inequality

Let $G$ be a finite group, $S \subset G$ a generating set. Set $\sigma(G):=\sum_{U \subset G} |U| $, where the sum runs over all subgroups $U$ of $G$. Set $H_G := \sum_{g \in G} \frac{1}{|g|+1}$, ...
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35 views

Do the properties defining the Selberg class imply the distribution of real parts of non trivial zeros of an L-function is strongly unimodal?

Selberg defined what is now known as the Selberg class as a class of L-functions fulfilling for essential properties, which are analyticity, Euler product, functional equation and Ramanujan-Patersson ...
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85 views

The Riemann Hypothesis follows from the Polya-Vinogradov inequality?

The Mertens function $M(x)$ is defined as $\Big|\sum_{n\leq x} \mu(n)\Big|$, where $\mu$ denotes the Mobius function. If $\chi$ is a primitive character modulo $q$, the Polya-Vinogradov inequality ...
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47 views

If I have a polynom $P(x)$, which zeros have the absolute value $q^{-(\frac{n-1}{2})}$. Why is this an accord to the Riemann hypothesis?

You can read the question above. So I'm really " new in terms of Riemann hypothesis". I have read about the hypothesis in wikipedia. So I know the statement of the Hypothesis : The Riemann Zeta ...
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104 views

What makes Riemann hypothesis so much harder to prove than its analogue for curves over finite fields

The analogue of the Riemann hypothesis for curves over finite fields has been shown by André Weil (see also Roadmap to Riemann hypothesis for curves over finite fields) and further deep results (Weil ...
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54 views

A probabilistic attempt to solve Riemann Hypothesis using Mertens function.

I know that the following statement: For every $\epsilon>0$ $$M(N)=O(N^{0.5+\epsilon})$$ is equivalent to Riemann Hypothesis (Where $M(N)$ is Mertens function). As Mertens function behaves somehow ...
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88 views

On an Inequality for the Riemann Zeta Function

Okay, firstly a bit of background to set the scene. My question comes from the approaches made by R. Spira in his paper, "An inequality for the riemann zeta function," regarding the initial steps he ...
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61 views

Reference for $|\pi(x) - Li(x)| \ll x^{1/2+\epsilon} \Rightarrow $ RH

It is a well-known result that the Riemann hypothesis is equivalent to the estimate $|\pi(x) - Li(x)| \ll x^{1/2+\epsilon}$. The forward implication was proved by von Koch all the way back in 1901. Is ...