# 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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### Breakthrough on zero density estimates on Riemann hypothesis

Terence Tao announced a breakthrough on Riemann hypothesis Original paper by Guth and Maynard. Tao writes: Let $N(\sigma,T)$ denote the number of zeros of the Riemann zeta function with real part at ...
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### Applications of the de Bruijn-Newman constant outside of the Riemann Hypothesis

According to Wikipedia, The de Bruijn–Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain ...
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### How do we test Riemann zeta zeros for simplicity?

I understand that we call a “simple zero” if the first derivative of the complex function $\neq 0$. How does this apply to the zeros of the Riemann zeta? I read that all known zeros are simple. Taking ...
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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 ...
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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 ...
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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? ...
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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 ...
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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: ...
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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 ...
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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 ...
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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'...
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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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### 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 ...
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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 ...
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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 ...