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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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Symmetry of zeros in the critical strip for Riemann Hypothesis

If it can be proven that there are no zeros for real values greater than $0.5$ in the critical strip, does this prove that there are no zeros in the critical strip having a real value of less than $0....
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28 views

Bound for the number of roots $\rho$ of $\xi(\rho)$

I was reading the book Riemann's Zeta Function, by H. M. Edwards, page 42, where is a theorem that estimates the number of roots of the $\xi$ function $$\xi(s)=\Gamma\Big(\frac{s}{2}+1\Big)(s-1)\pi^{-...
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If I have a polynom $P(x)$, which zeros have the absolute value $q^{-(\frac{n-1}{2})}$. Why is this an accord to the Riemann hypothesis?

You can read the question above. So I'm really " new in terms of Riemann hypothesis". I have read about the hypothesis in wikipedia. So I know the statement of the Hypothesis : The Riemann Zeta ...
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64 views

What makes Riemann hypothesis so much harder to prove than its analogue for curves over finite fields

The analogue of the Riemann hypothesis for curves over finite fields has been shown by André Weil (see also Roadmap to Riemann hypothesis for curves over finite fields) and further deep results (Weil ...
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A probabilistic attempt to solve Riemann Hypothesis using Mertens function.

I know that the following statement: For every $\epsilon>0$ $$M(N)=O(N^{0.5+\epsilon})$$ is equivalent to Riemann Hypothesis (Where $M(N)$ is Mertens function). As Mertens function behaves somehow ...
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1answer
466 views

Are there known zeros of the Zeta function off the line 1/2?

I've been doing research in information theory that had some superficial connections to the Zeta function, since I was making use of logarithms of complex numbers. In short, we'd be looking at the ...
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128 views

Is this equivalent to the Riemann Hypothesis?

By a result of Spira, we know that the Riemann Hypothesis (RH) is equivalent to the statement that $|\zeta(1-s)|$ increases as $\Re(s)$ varies on $(\frac{1}{2}, \infty)$ with $|t|=|\Im(s)|\geq 165$ ...
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1answer
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Why is the Euler product expected to play a role in a solution of the Riemann Hypothesis?

The Riemann Hypothesis is the statement that the Riemann zeta function $\zeta(s)$ does not vanish for $1/2<\Re(s)<1$. $\zeta(s)$ can also be expressed by the Euler product over primes $$\zeta(s)=...
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1answer
886 views

A line integral involving $\log \zeta(s)$

Let $\zeta$ denote the Riemann zeta function. Using the Cauchy integral theorem, can you evaluate $$I=\int_{\Re(s)=\frac{1}{2}} \frac{(2s-1)}{s^{2}(1-s)^2}\Bigg[\int \log((s-1) \zeta(s)) \mathrm{d}s\...
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An equation involving Non-Trivial Zeros of the Riemann Zeta function

$\rho$ is a Non-Trivial Zero of the Riemann Zeta function if and only if $$\displaystyle\int_1^{+\infty} \lfloor x\rfloor x^{-2-\rho} dx =\int_1^{+\infty} \lfloor x\rfloor \{ x \}x^{-2-\rho} dx $$ ...
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On an Inequality for the Riemann Zeta Function

Okay, firstly a bit of background to set the scene. My question comes from the approaches made by R. Spira in his paper, "An inequality for the riemann zeta function," regarding the initial steps he ...
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A computation in Conrey's paper on Riemann zeta function

I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here). I have question/doubt in a particular step: In P.10, it claimed ...
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Is there an analog of Robin's Inequality for L-functions?

In his 1984 paper Guy Robin showed the Riemann Hypothesis is equivalent to $\sigma(n)\lt e^\gamma n\log\log n$ for all integers $\gt$ 5040. The Riemann Zeta Function is a special type of L-function ...
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Questions related to the Riemann zeta function where $|\zeta(s)|=|\zeta(1-s)|$

The Riemann Zeta functional equation is defined as follows. (1) $\quad\zeta (s)=f(s)\,\zeta(1-s)\,,\quad f(s)=2^s\pi^{s-1}\sin\left(\frac{\pi\,s}{2}\right)\,\Gamma (1-s)$ Note that $|f(s)|=1$ along ...
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Is there an analogue of the Mertens function for the generalised Riemann conjecture

It is known that the Riemann conjecture is equivalent to $$M(x) = O(x^{\frac12+\epsilon}),$$ where M(x) is the Mertens function. Does there exist an analogue to this equivalence for the generalized ...
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Questions related to $f(x)$ where the Riemann Xi function $\xi(s)=s\int\limits_0^\infty f(x)\,x^{-s-1}\,dx$

I realize this question is a bit long and contains quite a few formulas, but I believe a considerable amount of background and context are needed to fully understand my questions below. Also, I ...
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Riemann hypothesis for $L(s,\chi)$ and $L(s,\chi^\sigma)$

If $\sigma \in \text{Gal}(\mathbb{Q}(\zeta_{\infty})/\mathbb{Q})$ do we know or expect that two Dirichlet L-functions $L(s,\chi)$ and $L(s,\chi^\sigma)$ have more in common, especially in term of ...
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Is comparing real and complex values within Robin's Inequality legal? And how would we?

I have a problem where I need to compare real and complex numbers. I see here and here that there are different ways to go about interpreting the sizes of complex numbers, but in my context I want to ...
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A function that satisfies $n^{f(n)}\zeta(f(n)) \to e^\gamma n\log\log n$

I have been trying to find a function $f$ that yields the following: $$n^{f(n)}\zeta(f(n)) \to e^\gamma n\log\log n$$ where $f(n)\to1$ and $f(n)\gt1$ for all sufficient $n$. I suspect that Mertens ...
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What is asymptotic and error bound for $\sum\limits_{k=1}^K\left(\frac{1}{\rho_k}+\frac{1}{\rho_{-k}}\right)$ as a function of $K$?

This is a follow-on of my previous question What is the convergence of the explicit formula for $\frac{\zeta'(s)}{\zeta(s)}$? My previous question was related to the following two formulas. (1) $\...
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What is the convergence of the explicit formula for $\frac{\zeta'(s)}{\zeta(s)}$?

The explicit formula for $\frac{\zeta'(s)}{\zeta(s)}$ illustrated in (3) below was derived from the relationship illustrated in (1) below using the explicit formula for $\psi(x)$ defined in (2) below. ...
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1answer
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Riemann Zeta function, nontrivial zeroes

How can we prove what are, say the first 4 non-trivial zeroes of the Riemann $\zeta$ on the critical line $Re(z_j)=\frac{1}{2}$, $j=1,2,3,4$ the first two with negative imaginary part and the second ...
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Questions related to Moebius Transform of Characteristic Function of the Primes

Consider the function defined in (1) below related to the fundamental prime counting function $\pi(x)$. Note that A143519(n) is not multiplicative. (1) $\quad f(x)=\sum\limits_{n=1}^{x}A143519(n)$ ...
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1answer
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What is the Todd's function in Atiyah's paper?

In terms of purported proof of Atiyah's Riemann Hypothesis, my question is what is the Todd function that seems to be very important in the proof of Riemann's Hypothesis?
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How is the Riemann Hypothesis related to P vs NP?

Are the Riemann Hypothesis and P vs NP related? It seems that if there is an algorithm to find the distribution of primes without factoring every number would be a polynomial time solution? I am ...
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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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1answer
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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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1answer
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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
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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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An upper bound to the number of divisors equivalent to Riemann hypothesis?

I have written a note, where I believe I can prove that the following upper bound on the number of divisors is equivalent to Riemann hypothesis: Let $\tau(n)$ be the number of divisors of $n$, $(n,l)$ ...
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1answer
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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
88 views

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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2answers
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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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1answer
276 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
429 views

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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2answers
63 views

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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0answers
165 views

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 ...