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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Z(t) having a positive local minima or a negative local maxima?

In H.M.Edwards' Riemann's Zeta Function, on page 176, he writes: "If there were a point at which the graph of $Z(t)$ came near to $Z = 0$ but did not actually cross it -that is, if $Z$ had a ...
sku's user avatar
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Is $\phi_0$ equivalent to the Riemann hypothesis?

This is an extension (and more distilled version) of Extension of PDE's to critical strip, with new information. I am fairly sure that my constructions are an alternate description of the De Brujn ...
John Zimmerman's user avatar
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Denjoy's Probabilistic Interpretation

Does Denjoy's Probabilistic Interpretation actually "prove" that the Mertens function ratio between numbers with odd number of distinct prime factors and even number of prime factors is 1? ...
NCY's user avatar
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Liouville Lambda Function and Riemann Hypothesis

What is the exact statement involving the Liouville Lambda function, which is equivalent to Riemann Hypothesis, and true iff RH is true? Can anyone cite the sources for it and/or outline its proof in ...
Ok-Virus2237's user avatar
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Using Model-Theoretic Proof of Ax-Grothendieck for the Riemann Hypothesis

A proof of Ax-Grothendieck utilizes model theory and the fact that the theorem is true for finite fields, and also algebraic closures of finite fields. See here. I have a (perhaps naive) question: ...
abiteofdata's user avatar
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Confirmation of Equivalent Form of Riemann Hypothesis

Can anyone, who has knowledge of the following, share some more details about it because not much information is available publicly regarding the same: RH is equivalent to the assertion that for all $...
Ok-Virus2237's user avatar
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Riemann Hypothesis follows from the statement $M\left(x\right)=o_x(x^{\frac{1}{2}+\varepsilon})$

Recall that the Mertens function is defined via: $$M(n):=\sum_{n\ge x\ge 1} \mu(x)$$ Where $\mu$ is the Möbius function. Littlewood proved that if $M\left(x\right)=o_x(x^{\frac{1}{2}+\varepsilon})$ ...
linuxbeginner's user avatar
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Winding of $\frac{\zeta(\sigma+it)}{\zeta^{\prime}(\sigma+it)}$ for $0 \lt \sigma \le \frac{1}{2}$

The image of the imaginary-positive critical line $\frac{1}{2}+it, t \ge 0$ in the function $ዘ(z) = \frac{\zeta(z)}{\zeta^{\prime}(z)}$ (i.e. the reciprocal of the logarithmic derivative of $\zeta$) ...
Szczepan Hołyszewski's user avatar
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Question about Salem's integral equation reformulation of Riemann hypothesis

Consider an integral equation: $$\int_{-\infty}^{+\infty}\frac{e^{-\sigma y}f(y)}{e^{e^{x-y}}+1}dy=0$$, where $\sigma\in(\frac{1}{2},1)$ Salem proved that this equation has no bounded solution other ...
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number of n.t. zeros in an interval in the limit

Motivation: We know that the density of non-trivial zeros of $\zeta(s)$ increases with $t$. But how dense does it get? Assuming RH. Summary Trying to solve for number of zeros in the length (measured ...
sku's user avatar
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Proof of PNT using RH

Is there any proof of Prime Number Theorem that follows immediately from the Riemann Hypothesis? I know that if RH is true then $\sum_{n \leq x} \mu(n) = x^{1/2 + \epsilon}$ for every $\epsilon > 0$...
Epsilon-Delta's user avatar
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Better bounds in the error term of the summatory function of Von-Mangoldt function and the Riemann Hypothesis

Theorem Let $f : \mathbb{N} \to \mathbb{C}$ be an arithmetic function and let $M(f, x) = \sum_{n \leq x} f(n)$ be the summatory function of $f$. If $M(f, x) = Ax^{\alpha} + O(x^{\theta})$, where $\...
Epsilon-Delta's user avatar
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Is there a link between Group Theory and the Riemann Hypothesis? [closed]

My question is twofold: Does anyone know if there's a connection between the Monster Group and the Riemann Zeta function? If there is a known connection, then I would be curious to know when and if ...
emery's user avatar
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Why doesn't the Riemann Zeta Function have zeroes at positive even integers? [closed]

According to the Wikipedia entry on "Riemann Functional Equation", the Zeta Function is equal to itself multiplied by a bunch of stuff, including the term $$\sin(πs/2)$$ This sine term means ...
Alexandra's user avatar
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Meaning of $M(n)=O\left(x^{\frac{1}{2}+\epsilon}\right)$

I am trying to fully understand the implications of $M(n)=O\left(n^{\frac{1}{2}+\epsilon}\right)$, where $M(n)$ is Mertens function, being equivalent to Riemann Hypothesis. (i) Is the equivalence ...
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Question concerning an assertion regarding the modulus of the Riemann Zeta function (follow up)

Update December 2023 -- Some additional cleanup, however the general argument remains the same. I have yet to see these telescoping/collapsing equations in the literature. Do reach out via email if ...
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Extraordinary Numbers

Can you please explain what are Extraordinary Numbers in detail? At the same time, I would also like to confirm whether the equivalent problem of Riemann Hypothesis mentioned here is correct (like it'...
Ok-Virus2237's user avatar
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Properties of sum appearing in Riemann's explicit formula

Let $R(x)$ be Riemann's function defined as $R(x) = \sum_{k=1}^\infty\frac{\mu(k)\text{li}(x^{1/k})}{k}$ where $\mu$ is the Moebius function and li the logarithmic integral. Let $\pi(x)$ be the prime ...
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Is this inequality related to $|\log \zeta (s)|$ wrong?

This is a doubt in the Littlewood's first estimate(in which it is assumed that Riemann hypothesis is true) given in this book, page 433. Let $s = \sigma + it$ and $\delta $ such that $\frac12 + \...
Eloon_Mask_P's user avatar
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Logarithmic Integral evaluation

I was watching this video explaining the Riemann hypothesis, and there is the use of the Li(x), Logarithmic integral function: https://youtu.be/GEcHgadgOn4 It seems that this integral cannot be ...
Thomas Moore's user avatar
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Are there any "Lehmer triplets" among the zeros of Riemann zeta function?

In the set of non-trivial zeros of the Riemann zeta function, there are some numbers that are unusually close to each other. These are called Lehmer pairs, the first of which appears at the 6709th and ...
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Where are the zeros of a slightly perturbed Riemann Zeta function?

We seek to understand the locations of the zeros when we introduce a minor perturbation to the Riemann Zeta function: ${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}...
david's user avatar
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Landau-Siegel zeros: Why can't they be found?

Let me situate myself in this discussion: I'm not a mathematician or someone who had a comprehensive mathematical training. Yet I do have a great interest in Mathematics and I've been recently reading ...
Carlos Gouveia's user avatar
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Does $\sum_\rho R(x^\rho) \sim \sqrt{x}/\log(x)$ assume the Riemann hypothesis?

I am learning about the exact formula for the prime counting function $\pi(x)=R(x)-\sum_{\rho}R(x^\rho)$ where $R$ is Riemann's R-function $R(x)=\sum_{k=1}^\infty\frac{\mu(k)}{k}li(x^{1/k})$, $li$ the ...
amanwithnoname's user avatar
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Are the imaginary parts of these Riemann zeta related numbers equal?

While we can't say anything about the real part of these numbers below, maybe it could be possible to say something about the imaginary parts? Let: $$A=\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{...
Mats Granvik's user avatar
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Is this prime number hypothesis correct ? And if it isn't what would it contribute to math if it was?

I have an hypothesis and I tried a lot of prime numbers on it and it worked , so if this hypothesis is correct ; How can it contribute to math ? Let $y$ be a non-zero natural number; Let $p_y$ be ...
Ali Himeur's user avatar
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Spectrum of "infinite-Gram matrix"?

This question comes out of my interest for positive definite kernels over the natural numbers. (I have collected some kernels with proofs). First let me point to a connection between self-adjoint ...
mathoverflowUser's user avatar
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Zero Free Region of Riemann Zeta Function for the line $\Re(s)>1-\epsilon$

I am familiar with the classical zero-free region of $\zeta(s)$. That is if $\rho=\sigma+it$ is a non-trivial zero of the Riemann zeta function the classical zero-free region says that there are no ...
Steven Creech's user avatar
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Methods Of Counting Non-trivial Zeros Of Zeta Function

What are the different methods using contour integration that can be used to count the non-trivial zeros of the zeta function? I know of Backlund's exact formula which is similar to Riemann-von ...
Yash bodhi's user avatar
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If we calculate the Riemann Zeta function along a vertical line in the critical strip, for Re(z)!=0.5, do we ever hit the same point 3 times or more

The line $\zeta(0.5+x \mathrm{i})$ (for $x$ real $>0$) hits the point $0+0i$ an infinite number of times. Along a different line, I'm interested not in hitting zero, but hitting any point more than ...
Michael Lachmann's user avatar
3 votes
2 answers
194 views

RH as asymptotic order of Liouville’s partial sum function

E. Landau famously proved in his 17 page PhD thesis from 1899 that $$ \lim_{n\to\infty}\frac{\sum_{k=1}^{n} \lambda(k)}{n^{\frac{1}{2} +\varepsilon}} =0, \; \forall \varepsilon>0$$ for the ...
Raphael J.F. Berger's user avatar
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How To Use Complex Contour Integration To Obtain Zeros Of Riemann Zeta Function

Specifically, can someone recommend resources that continue from the end of this YouTube video: https://www.youtube.com/watch?v=uKqC5uHjE4g&t=2s&ab_channel=zetamath? Additionally, are there ...
Yash bodhi's user avatar
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Riemann Hypothesis implies Prime number theorem?

Following this thread Prime number theorem and Möbius μ function I was looking into Apostol's book, Introduction to Analytic Number Theory, Springer 2000. And also note that RH is equivalent to ...
linuxbeginner's user avatar
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Number of zeros in the critical strip and complex argument

According to the Wikipedia article on Riemann hypothesis, The number of zeros of the zeta function with imaginary part between $0$ and $T$ is given by $$N(T)=\frac{1}{\pi}\operatorname{Arg}\left(\xi\...
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Argument principle and Riemann hypothesis

Argument principle states: If $f$ is a meromorphic function inside and on some closed contour $C$, and $f$ has no zeros or poles on $C$, then $$\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}\,dz=Z-P$$ ...
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Riemann Hypothesis Research

I don't know if this is the correct place for this. If not feel free to remove! I am a recent graduate of a BSc in Applied and Computational Maths and am now not working in a maths field. I miss ...
Barry O'Keeffe's user avatar
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174 views

Why is the Riemann explicit formula for primes (via the Riemann Hypothesis) any better than existing formulae?

Many people, such as here, don't consider Willan's formula for primes, and other such formulas given here, as meaningful formulae for computing primes. My understanding of the main criticisms are that ...
Tanishq Kumar's user avatar
10 votes
1 answer
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Why do number theorists care so much about how well $\text{Li}(x)$ approximates $\pi(x)$ if it's not our best approximation?

An alleged primary motivator for the RH is so that we can bound the error term $|\text{Li}(x) - \pi(x)|$ by a factor of $O(\sqrt{x}\log x)$. However, I also learned about Riemann's explicit formula $R(...
Tanishq Kumar's user avatar
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Formula for Mertens Function $M\left(x\right)$ in terms of the Prime counting function $\pi\left(x\right)$ using inclusion-exclusion

In this post I was interested in studying the behaviour of Mertens function $M\left(n\right)$, which is defined for all positive integers as $$M\left(n\right)=\sum_{k=1}^{n}\mu\left(k\right)$$ Less ...
Juan Moreno's user avatar
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Critical strip...critical cuboid?

If you know about the Riemann hypothesis then you probably are aware of the critical strip. In short, I'm wondering how to "reverse-construct" a function based on the generalization of the ...
John Zimmerman's user avatar
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2 answers
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What is the error in the following reasoning regarding the Riemann Hypothesis?

I encountered the following earlier today and was wondering where, or if there are, errors in the reasoning. Assume the Riemann Hypothesis is independent of ZFC. Assert that the RH is false, though ...
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1 vote
1 answer
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An unusual equivalent form of Riemann hypothesis

Let $G(x)=\sum_{k\leq x}\frac{\mu(k)}{k}$, where $\mu$ is the Mobius function. From this question and its answer, its mention the Riemann hypothesis is equivalent to $G(x)=O(x^{-\frac{1}{2}+\epsilon})$...
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Reference about equivalent form of the Riemann hypothesis

I saw a statement about the Riemann hypothesis in Wikipedia, stating the following: $\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}=\frac{1}{\zeta(s)}$ holds for $Re(s)>\frac{1}{2}$ is equivalent to the ...
Ken.Wong's user avatar
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Where can I learn about the Riemann Zeta function/Riemann Hyposthesis?

I've been very curious about the Riemann zeta function and the Riemann hypothesis and I have been wanting to learn about it on a deeper level for a while now. While I don't intend to understand ...
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Why was the number $73.2$ used in "Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II" in theorem 10, inequality 6.2?

In Schoenfeld's paper "Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II," theorem 10, inequality (6.2) states "If the Riemann hypothesis holds, then $|\psi(x) - ...
mathlander's user avatar
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Turán proof that constant sign of Liouville function implies RH

In Mat.-Fys. Medd. XXIV (1948) Paul Turán gives what he says is a proof of the statement that if the summatory $L(x) = \sum_{n\leq x} \lambda(n)$ of the Liouville function $\lambda(n) = (-1)^{\Omega(n)...
Tommy R. Jensen's user avatar
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What is the function E(x)?

I’m an electrical engineering student getting a minor in math. I recently started my first pure math class and my professor proposed something interesting on the board. I don’t remember exactly what ...
Nasri Ibrahim's user avatar
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Proof that $\int_0^\infty\ln^{s-1}(t+1)dt=\int_0^\infty\frac{\ln^{s-1}(t+1)}{t}dx$

I proved that$$\int_0^\infty\ln^{s-1}(t+1)dt=\int_0^\infty\frac{\ln^{s-1}(t+1)}{t}dt.$$ When $s$ is a zero of the zeta function $\zeta(s)$. Is my proof correct?:$$0=\int_0^\infty\frac{t^{s-1}}{e^t-1}...
Kamal Saleh's user avatar
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4 votes
2 answers
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Is there an absolute mimumum distance between zeros of the Riemann zeta function?

I believe that the nontrivial zeros of the Riemann zeta function get increasingly spaced out the further they are from the real axis. This suggests that there is probably an absolute minimum distance ...
tparker's user avatar
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On the zeros of the Riemann Zeta function

Let us consider the infinite set of non-trivial zeros of Riemann zeta function $\{\rho_n \}$ and the following product $$ \prod_{n=1}^{\infty}\bigg(1-\frac{1}{\rho_n}\bigg)\bigg(1-\frac{1}{\overline{\...
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