Questions tagged [riemann-hypothesis]

Question 1

Riemann's Zeta Function Page 17: note $\psi(x)=\sum_{n=1}^{+\infty}e^{-n^{2}\pi x}$ \begin{equation} \xi(1/2+ti)=4\int_{1}^{+\infty}\frac{d[x^{3/2}\psi^{\prime}(x)]}{dx}x^{-1/4} \cos\left(\frac{t\ln x}...

Question 2

This is related to P. Deligne's paper La conjecture de Weil I, (5.8 b) (pp. 293) and the proof of (7.1) (p. 300). In (5.8) the author shows that there is a short exact sequence \begin{equation*} 0\...

Question 3

Let $$\pi(x)=\sum\limits_{p\le x} 1\tag{1}$$ be the prime-counting function and $$\rho=\alpha+i \gamma\tag{2}$$ represent a non-trivial zeta zero in the critical strip. This question is about formula ...

Question 4

I've seen it reported that we've found all of the nontrivial zeros to the Riemann zeta function up to some large height, and that all of them have real part $1/2$; i.e. a counterexample would need ...

Question 5

Let $s=\frac12+iz$ be a non-trivial zero of the Riemann Zeta function $\zeta(s)$. Let $f(s,a)=\frac12+iaz$ where $s=\frac12+iz$ be the $a$ complex conjugate of $s$. Eg: $f(s,1)=s$ and $f(s,-1)=\...

Question 6

Let $s=\frac12+iz$ be a non-trivial zero of the Riemann Zeta function $\zeta(s)$. Would the complex conjugate $\overline s$ satisfy $\zeta(\overline s)=0$? If not generally would they ever satisfy the ...

Question 7

First of all, please note that I'm just a curious amateur, and the goal of this question is purely out of curiosity rather than directed research. For a reason I don't really understand, it seems ...

Question 8

I am having difficulties into understanding the last part of the proof of Hasse Corollary of the Riemann hypothesis over finite fields: Let $g \in \mathbb{Z}[X]$ be a cubic polynomial with no multiple ...

Question 9

I came to know that $$\frac{\pi\left(x\right)-\text{Li}\left(x\right)}{\frac{\sqrt{x}}{\log x}}=-1-\sum _{\ \zeta \left(\frac{1}{2}+i\gamma \right)=0}^{ }\frac{x^{i\gamma }}{\frac{1}{2}+i\gamma }+O\...

Question 10

I have been studying the Riemann Hypothesis for some time and I have recently stumbled upon the book "The Riemann Hypothesis A Resource for the Afficionado and Virtuoso Alike" (https://link....

Question 11

We know, assuming eta function, that $$\eta(x) =\sum_{n=1}^{\infty}(-1)^{1-n}\frac{1}{n^{x}}$$ and $$\eta (x)=(1-2^{1-x})\zeta (x)$$ so we have zeros of this function at $x=1$ for $$\frac{2\pi k}{\ln\...

Question 12

For example, Riemann hypothesis is absolute and that's why we can't apply forcing to obtain independence result. But is negation of Riemann hypothesis also absolute? That is, if ¬RH is true in ZFC, ...

Question 13

I’m honestly in a bit of shock after coming across this fascinating rule from the Clay Mathematics Institute's official problem guidelines: "If, alternatively, the counterexample shows that the ...

Question 14

After reading it I have no clear idea if proof from Guy Robin "Sur l’ordre maximum de la fonction somme des diviseurs" or in other similars related to Merten's Theorems (2nd and 3rd) does ...

Question 15

Let $(X_i)_{i \in \Bbb N}$ be a sequence of independent RV's with zero means $\Bbb E[X_i]=0$ and let $(b_i)_{i \in \Bbb N}$ be an increasing sequence of positive numbers satisfying $\sum_i \Bbb E[X_i^...

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Questions tagged [riemann-hypothesis]

Convergence of Integral Expressions for ξ(s) Involving ψ'(x) and ψ''(x) in the Complex Plane

A question related to the Riemann hypothesis in positive characteristic: Why does $H^{n+1}(X_{u},\mathbb{Q}_{\ell})$ occur in the long exact sequence?

Reference request on a formula proposed by Riemann relating the prime-counting function $\pi(x)$ to the non-trivial zeros of $\zeta(s)$

How do we know we're finding all of the zeros to Riemann zeta function?

Scaled complex conjugates of non-trivial zeros of Riemann Zeta function

Complex conjugates of non-trivial zeros of Riemann Zeta function [closed]

About a curious identity for the Mertens function

A Corollary of the Riemann Hypothesis for one-variable multiplicative sums (A. Weil)

An estimate of $\pi(x)$ assuming RH [closed]

Equivalence of the Riemann Hypothesis

Symmetry line on zeta function [closed]

Absoluteness in logic also applies to its negation?

Can the Riemann Hypothesis Survive a Counterexample? [closed]

Does proof on alternating sign error term of Mertens theorems depend on RH?

Quantitative version of law of large numbers and application to RH

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