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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Proof by contradiction and ZF set theory

The following is quoted from Sir Michael Atiyah's draft proof of Riemann Hypothesis, Section 5, stating that 'To be explicit, the proof of RH in this paper is by contradiction and this is not accepted ...
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Theorem 14.25(A) in Titchmarsh “The theory of the Riemann zeta-function”

In Titchmarsh's book "The theory of the Riemann zeta-function" there's theorem 14.25(A) on page 369 of the second edition where a summand $1/\zeta(s)$ appears out of the blue, so it seems... Oh, I do ...
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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 ...
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Is there an analytic function with zeroes only at $-2n$, and zeroes at $\frac12\pm it$, and further, symmetric zeroes within the critical strip?

Is there an analytic function with zeroes only at: every $-2n$, $\frac12\pm it$, and at least one at $\frac12\pm\epsilon\pm it$ where $0<\epsilon<\frac12, t\neq0$ (and these zeroes observing ...
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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 $...
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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 ...
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What does the Lindelöf hypothesis imply?

I recently read this article (https://viterbischool.usc.edu/news/2018/06/mathematician-m-d-solves-one-of-the-greatest-open-problems-in-the-history-of-mathematics/) about someone who may have proven ...
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What are the current “records” for Robin's criterion?

Define the function $f:\mathbb{N}\to\mathbb{R}$ as $$ f(n)=e^\gamma n \log \log n, $$ where $e$ is Euler's constant and $\gamma$ is the Euler-Mascheroni constant. Then Robin's criterion states that ...
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1answer
192 views

Reciprocal of Riemann zeta function and Moebius function

According to Wikipedia, the Dirichlet series $$\sum^\infty_{n=1}\frac{a_n}{n^s}$$ converges absolutely for $Re(s)>k+1$ where $a_n$ is $O(n^k)$. For the reciprocal of Riemann zeta function $\frac1{\...
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Unclear multiplication with Riemann zeta functional equation with eta

In the region 0 < Re(s) < 1 we know that $$ \zeta(s) = 1/(1-2^{1-s}) \sum_1^\infty (-1)^{n+1}/n^s\,. $$ This is a multiplication of two complex numbers. Question 1: Am I right to suppose ...
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Clarrification Regarding the Robin Inequality

I just read a paper related to the Robin Inequality, and the abstract read: "Abstract. Let σ(n) denote the sum of divisors function. In this paper we give a simple proof of the Robin inequality (R): ...
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Questions and concerns

I would like to know if solving the riemann hypothesis as well as the twin prime conjecture are still questions within mathematics? I have been unable to find credible resources.
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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 $\...
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Shrinking circle on Zeta zeroes. $\zeta(s + \Delta r e^{i \theta})$ when $\zeta(s) = 0$ and $\Delta r \rightarrow 0^{+}$.

I have a very simple question about the Riemann Hypothesis that's probably quite obvious to somebody with more experience in complex analysis. I was having trouble with the same type of problem in my ...
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1answer
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Question on Proof of the Equivalence of two Coefficient Functions Related to the Dirichlet Series for $\frac{\zeta(s+1)}{\zeta(s)}$

I derived the relationship illustrated in (1) below which I believe converges for $s>0\lor\Re(s)>\frac{1}{2}$ assuming the Riemann hypothesis. The function $rad(n)$ is the radical or square-free ...
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Textbooks for studying Riemann hypothesis

I'm a physics graduate recently learned Riemann hypothesis in a mathematical physics course. ( I knew what the hypothesis is but didn't know mathematical statement) I got interested, and I wanna ...
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396 views

What theorems depend on the unproven Riemann Hypothesis?

I heard that many theorems in modern mathematics depend on the unproven Riemann Hypothesis. What are some examples of such theorems, and what will be the impact on these results if the Riemann ...
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1answer
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How much of the Riemann Hypothesis has been solved?

From Wikipedia, I read ...the Riemann Hypothesis is a conjecture that the Riemann Zeta function has its only zeroes at the negative even integers and complex numbers with real part $\frac{1}{2}$. ...
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What is upper bound for the largest prime in a counter-example for robin's inequality

What is upper bound for the largest prime in a counter-example for robin's inequality ?! Assume that $\frac{\sigma(n)}{ n \ln \ln n} > e^\gamma$ for some $n>5040$, and let $p$ be the biggest ...
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Questions on Prime Counting Functions, Explicit Formulas, and Related Zeta Functions

This question is related to the prime-counting functions defined in (1) to (4) below. (1) $\quad A(x)=\sum\limits_{n=2}^\infty\frac{\Lambda(n)}{\log(n)^2}\,\theta(x-n)$ (2) $\quad\Pi(x)=\sum\limits_{...
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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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1answer
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How to locate zeros of the Riemann Zeta function?

I'm trying to locate the first 1000 zeros of $\zeta(s)$ and not sure about the best way to go about it. I was considering the Newton-Raphson method but can'd find a good way to code $\zeta'(s)$ in ...
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Heuristics on the asymptotic behaviour of the divisor funcion

Let $\sigma(n)$ be the sum of divisor function. We know, by Gronwall's Theorem, that $${\lim \text{sup}}_{n \to \infty} \frac{\sigma(n)}{n \log \log{n}} =e^\gamma$$ And the Riemann Hypothesis (...
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contributions of Riemann Hypothesis to physics if the Riemann zeta function is a solution for known differential equation? [closed]

There are several consequences of the Riemann hypothesis in many area as Number theory , complex analysis $\cdots $ ,I'm interesting to know what about those consequences if the Riemann zeta function ...
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How to contour integrate the Riemann Zeta function with a goal to verify the Riemann hypothesis?

I've counted the number of zeros of $\zeta$ by checking sign changes of $Z(t)$. I used a tolerance of 0.1 and found 649 zeros less than 1000, and then a tolerance of 0.05 to find 10,142 zeros less ...
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Reference for von Koch's 1901 theorem (RH characterization)

In Stein and Shakarchi's Complex Analysis, chapter 7, p.204, they give the following problem. [Problem 4] Show that $$ \pi(x) - \text{Li}(x) = \mathcal{O}(x^{\alpha+\epsilon}) \quad \text{ as } x ...
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1answer
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Verify the Riemann Hypothesis for first 1000 zeros.

I'm trying to verify the Riemann hypothesis for the first 1000 zeros using the Euler-MacLaurin expansion of $\zeta(s)$. But here's where my problems begin, firstly my python code doesn't seem to be ...
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772 views

Is this a spam message or the Riemann hypothesis really has been proved? [closed]

I have received now message from so@one-zero.eu include paper show that the Auther proved the Riemann hypothesis as shown here in this paperwhich include both relation between the distribution of the ...
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Riemann explicit formula for $\psi^*(x)$ in the region $0\leq x\leq 1$.

Is it possible to extend this Riemann explicit formula to interval $0\leq x\leq 1$? $$\psi^*(x)=x-\sum_{\rho} \frac {x^{\rho}}{\rho}-\frac{\zeta'(0)}{\zeta(0)}$$ Sum over trivial zeros of zeta $\sum ...
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Does knowing the associated determinant show that the Hilbert-Polya conjecture is meaningless, or is this approach too naive?

Let $A$ be the matrix: $$A = \text{If } n \bmod k=0 \text{ then } 1\text{ else }0$$ Let $B$ be the matrix: $$B=\text{If }k \bmod n=0\text{ then }\Im(\rho _n)\text{ else }0$$ where $\Im(\rho _n)$ ...
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Riemann hypothesis and the logarithmic integral

As it is stated, for instance, in Wikipedia, the Riemann hypothesis is equivalent to $$ |\pi(x)-{\rm li}(x)|< \frac1{8\pi}\sqrt x\log x,\qquad \mbox{for all } x\geq 2\,657, $$ but "li" denotes ...
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Is the Generalized Riemann Hypothesis thought to be true?

The Riemann Zeta Function is the analytic continuation of the following function: $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$ The Riemann Hypothesis states that the zeros of this in the critical ...
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Riemann explicit formula for $\pi(x)$ and its evaluation

We have for $x>1$: $$\tag{1}\pi^*(x)=R(x)-\sum_{\rho} R(x^{\rho})$$ $\rho$-s are zeros of Riemann zeta function (trivial and nontrivial). I know how to derive $(1)$ and also know how to derive: ...
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What are the allowed axioms to solve the Riemann Hypothesis?

First, is it even possible to state a well known problem and know what axioms are allowed for it? I'd guess that one must look at the branches of math involved, but I'm no mathematician. Therefore, I ...
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Meaning of Riemann R-function

Riemann R-function is defined as: $$R (x)=\sum _{n=1}^{\infty } \frac{\mu (n) \text{li}\left(x^{1/n}\right)}{n}$$ I know how it appears in Riemann explicit formula for prime counting function $\pi (x)...
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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,...
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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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1answer
64 views

zeros of a complex function defined by integers

Is there a known increasing sequence of positive integers $\{\textbf{a}\} = a_0<a_1<a_2<.....$ such that all the zeros $z_k$ on $\Re[z]>0$ of the complex function $F(z;\{\textbf{a}\})= \...
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Permutation of natural numbers relative to the Mobius Function

We can define a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that: $f$ is bijective. If $f(n)=m$, then $f(m)=n$. If $f(n)=m$ then $\mu(n)+\mu(m)=0$ We want to define $f$ such that the ...
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Using linear algebra to study number theory?

I've posted a paper on arXiv that outlines a linear algebra approach to number theory. Specifically, I have the following questions: Is it possible to draw connections between the factorization ...
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is $\zeta(s) = \frac{1}{1-2^{1-s}}\eta(s)$ an analytic continuation of $\zeta(s)$ for $\sigma > 0$

It seems from what I have read on the net, that the above representation of $\zeta(s)$ is a valid analytic continuation of $\sum_{i=1}^{\infty}\frac{1}{i^s}$ for $\sigma > 0$ except for a simple ...
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Riemann hypothesis and Robin's Inequality Implications

Is the following statement true: Let $\Bbb{A}$ be the set of all Natural numbers n, greater than or equal to 5041, for which the inequality $\displaystyle \sigma(n)<e^{\gamma}n\log\log n$ is not ...
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What is meant by the term “discrete” number with respect to the imaginary part of the non-trivial zeros of $\zeta(s)$?

The following link indicates the imaginary parts of the non-trivial zeros of the Riemann zeta function $\zeta(s)$ are "discrete" numbers. New Insight into Proving the Riemann Hypothesis What is ...
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Riemann Hypothesis numeric verification question?

As I found in wikipedia Riemann Hypothesys has been verified numerically by X. Gourdon (2004) up to 10000000000000 ($10^{13}$) zeroes. I have a few question about how they did it. I tried to read on ...
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An explanation of the importance of analytic formulas representating arithmetic functions related to equivalences to the Riemann Hypothesis

I'm curios about the following question, from an informational viewpoint. What is the purpose in finding/getting analytic formulas for specific arithmetic functions in the context of the Riemann ...
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question on Riemann $\zeta(s)$

I have a question that is troubling me. From the functional equation of $\zeta(s)$, can we not conclude that both $\zeta(s)$ and $\zeta(1-s)$ have the same non-trivial zeros (differing at most in ...
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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 ...
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Does statement (1) imply statement (2)? [closed]

Statement 1 : (Robin) proved that if the R.H. is false then there exist constants $0<\beta <\frac{1}{2}$ and $c>0$ small , such that $\sum \limits_{d|n} d \geq e^\gamma n \ln \ln n+ n\frac{ c ...
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148 views

Zeros of Riemann Zeta Function- Euler Product and Functional Equation

The problem statement, all variables and given/known data Question Use the functional equation to show that for : a) $k \in Z^+ $ that $ \zeta (-2k)=0$ b) Use the functional equation and the euler ...
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Riemann hypothesis, random walks and Möbius function

How to show that the Riemann hypothesis, random walks and the Möbius function are related or even equivalent? I was reading the paper Randomness and Pseudorandomness by Avi Wigderson, but to me the ...