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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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If Hardy proved that there are infinite zeros lying on the critical line, does not that mean he proved the Riemann Hypothesis? [closed]

This is an exerpt of the wiki page covering Riemann Hypothesis: *" Thus, if the hypothesis is correct, all the non-trivial zeros lie on the critical line consisting of the complex numbers + i t, ...
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Alternative expressions of the Riemann Siegel Z-function

In the Ivić, Aleksandar (2013). The theory of Hardy's Z-function, page 16 the Hardy function is defined as $$\begin{align} Z(t) &:= \zeta \left(\frac{1}{2}+it\right)\left(\chi\left(\frac{1}{2}+...
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1answer
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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) = \...
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Are the roots of the Riemann zeta function approximated through numerical algorithms?

Out of curiosity, I was taking a look at the R {pracma} package zeta() function, and specifically the example ...
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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 ...
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Prime Number Theorem/GRH versus estimations

Sometimes I believe elementary clues are lacking to me: First instance (PNT). I have a Schwartz function $\phi$, whose Fourier transform $\widehat{\phi}$ has support in $(-2,2)$. Why does prime ...
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2answers
74 views

Where to search for anomalous local extrema of Z(t) ?

Researchers such as Gourdon, who have computed huge numbers of zeros of zeta on the critical line, often list examples of “Lehmer’s phenomenon”, where a pair of zeros are so close together that the RH ...
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Can this integral giving numerically the first 126 Riemann zeta zeros be solved symbolically?

I imagine integration works a bit like a Russian doll where you have to open every doll/function in order to get to the core doll/function to do the actual integration on a power/Dirichlet series of $...
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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 ...
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1answer
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zeta function, a nontrivial zero that is not on critical line

The anylitical continued version of the zeta function is the eta function divided by $(1-2/2^s)$, since $(1-2/2^s)$ is never $\infty$, The zeros of $\zeta(s)$ the must be the same as the zeroes of $\...
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Riemann hypothesis: is Bender-Brody-Müller Hamiltonian a new line of attack?

There is a beautiful paper in Physical Review Letters [PRL 118, 130201 (2017), DOI:10.1103/PhysRevLett.118.130201] by Carl Bender, Dorje Brody, and Markus Müller (BBM) on a Hamiltonian approach to the ...
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Limit involving a sum over prime numbers and the logarithmic integral

Is it known wether the following limit tends to Infinity or not? Is there any possibility for it to converge to a constant? $$\lim_{n \to \infty} \sum^n_{p\le n} \frac{1}{\sqrt{p}} - li(\sqrt{n})$$ (...
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Questions on Convergence of $\int_1^\infty\left(\psi(x)-x\right)\ x^{-s-1}\ dx$ and $\int_1^\infty\left(\psi'(x)-1\right)\ x^{-s}\ dx$

Relationships (1) and (2) below are valid for $\Re(s)>1$. (1) $\quad\int_1^\infty\psi(x)\ x^{-s-1}\ dx=-\frac{\zeta'(s)}{s\ \zeta(s)}\,,\quad \Re(s)>1$ (2) $\quad\int_1^\infty\psi'(x)\ x^{-s}\ ...
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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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2answers
720 views

Why did Riemann believe that all non-trivial zeros of the zeta function lie on the critical line?

Bear with me, I'm fresh out of high school so my level of mathematical knowledge is quite low (probably too low to be trying to understand the Riemann hypothesis, but at least I'm trying). At this ...
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What is the explicit formula for the staircase function $S(x)$ defined in terms of the zeta zeros?

Riemann and von Mangoldt derived explicit formulas for the Riemann prime power counting function $J(x)$ and the second Chebyshev function $\psi(x)$ respectively via the following relationships. (1) $\...
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Questions on Convergence of $\int_{1-\epsilon}^\infty \left(\left(\sum_n \delta(x-n)\right)-1\right)\,x^{-s}\,dx$

Assuming the following definition for $G(s)$, the context of my questions below is the convergence of $Re(G(s))$ to $Re(\zeta(s))$ for $Re(s)\in(0,\,1)$. (1) $\quad G(s)=\int_{1-\epsilon}^N\left(\...
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1answer
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Would the falsity of RH imply the existence of a yet unknown functional equation? [closed]

The question is in the title : suppose RH is false. Would it imply that there exists a yet to be discovered functional equation relating the values of the Riemann zeta function at different points ...
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Questions on Implications of Riemann Hypothesis with Respect to $\psi\left(e^u\right)$

I've been told the Riemann hypothesis implies (1) below. (1)$\quad\psi\left(e^{\ u}\right)-e^{\ u}=o\left(e^{\ u\ (1/2+\epsilon)}\right)$ I've also been told (1) above implies (2) below converges ...
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Questions on Fourier Transform of $\frac{\psi[e^u]-e^u}{e^{u(1/2+\epsilon)}}$

In a response to one of my earlier questions which I believe was related to Evolution of Zeta Zeros from Fourier Transform of $e^{-t/2}\left(\psi'[e^t]-1\right)$, it was suggested I instead focus on ...
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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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The zeta function has infinitely many zeros in $0<\Re{s}<1/2$?

The following paragraph appears on page 42 in the book Rational Number Theory in the 20th Century: From PNT to FLT (Par Wladyslaw Narkiewicz): The fact that the strip $0<\Re{s}<1/2$ contains ...
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How is this elementary problem equivalent to the Riemann Hypothesis?

The problem I'm referring to is: Lagarias' Elementary Version of the Riemann Hypothesis, which states: For a positive integer $n$, let $\sigma(n)$ be the sum of all of its positive divisors. Let $...
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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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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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What is theoretical convergence of formula for $\zeta(s)$?

Assume the following definitions: $\zeta(s)$ - Riemann zeta function $\delta(x)$ - Dirac delta function My question is: What is the theoretical convergence of the following formula for $\zeta(s)$? ...
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Implications of a correct proof of the Riemann hypothesis? [duplicate]

What are some major implications of a proof of the Riemann hypothesis?
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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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Is there an equivalent statement of the Riemann hypothesis in quantum theory?

The question in the title. I know that there is a Hilbert–Pólya conjecture, but it is not equivalent. P.S. There is no quantum-theory tag.
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Possible Riemann's Hypothesis proof? [closed]

First of all, I imagine it will not be correct, just because of its simplicity, but I would also want to know why, as I can't find any mistake on it. The "proof" would be based on convining two main ...
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Weak Riemann Hypothesis?

The Riemann Hypothesis says that all non-trivial zeros of the Riemann zeta function lie on $Re(z)=\frac{1}{2}$ line instead this region $Re(z) \in (0,1)$. It seems a natural question to be that ...
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A Neat Identity Involving Zeta Zeroes

While playing around, I encountered the following very curious and cool identity. Consider the exponential integral $\text{Ei}(x)$ and the $n$th nontrivial zero of the Riemann Zeta function $p_n$. ...
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The Mellin transform for $\eta(s)\,\Gamma(s)$ and information about the distribution of primes.

The Mellin transform for: $$\displaystyle \Gamma(s)\,\zeta(s)= \int_0^\infty x^{s-1} \frac{1}{e^x-1}\,dx$$ equals: $$\frac{1}{2\,\pi\,i}\int_{c-\infty\,i}^{c-\infty\,i} s^{-x}\,\Gamma(x)\,\zeta(x) ...
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Equivalent to Riemann Hypothesis

Through last number theory, I did learn that Riemann hypothesis is equivalent to the following inequality : $|\pi(x)-Li(x)| \leq \sqrt{x} log(x)$ where $Li(x)$ is the Logarithmic integral function and ...
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Does Montgomery's pair correlation conjecture also hold true for Dedekind Zeta function ?

All: Does Montgomery's pair correlation conjecture also hold true for the zeros for Dedekind Zeta function of an algebraic number field K ? My understanding is that Montgomery's pair correlation ...
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A Formulation of The Riemann Hypothesis

To preface this inquiry, I am aware of the fastidious nature of the mathematics presented on this exchange. In addition to that, I apologize for my ineptness and the contention in this particular ...
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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 ...
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Proof (claimed) for Riemann hypothesis on ArXiv

Has anyone noticed the paper On the zeros of the zeta function and eigenvalue problems by M. R. Pistorius, available on ArXiv? The author claims a proof of RH, and also a growth condition on the ...
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“The music of the primes” by Sautoy: Is $\sqrt x$ correct?

I have recently read "The music of the primes" by Marcus du Sautoy (see excerpt here >>>). There he writes: "So how fair are the prime number dice? Mathematicians call a dice "fair" if the difference ...
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Finding high prime numbers assuming the Riemann hypothesis [closed]

Assuming the Riemann hypothesis is true, is it really possible to find high prime numbers more easily?
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What is the equivalent statement of GRH in term of Redheffer Matrix or Farey Sequences?

We all know that Riemann Hypothesis (RH) has many equivalent statements. There is one statement which expresses RH in term of Redheffer matrix, there is another equivalent statement of RH which ...
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The random matrix for Riemann Hypothesis, is it corresponding to an operator in quantum mechanics or in quantum field theory?

Odlyzko showed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of random matrices drawn from the Gaussian unitary ensemble. This ...
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How good is the pseudo-radom-sequence assuming the truth of RH?

The Riemann hypothesis is equivalent to the claim that the sequence of moebius-values (numbers not being squarefree are skipped) behave similar to a random walk. Let's assume that the Riemann ...
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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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If the value of Mertens function follow normal distribution, does this imply Riemann Hypothesis?

If the value of Mertens function follows normal distribution, does this imply Riemann Hypothesis ? I thought the answer shall be NO, because normal distribution still has "long tail".
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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} + ...
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Is a strong form of Goldbach conjecture equivalent of Generalized 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 ...
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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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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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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 ...