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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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Another One on the Riemann Hypothesis

I keep trying to ask the right questions about the Riemann Hypothesis in order to address my confusion but my understanding is so poor that I'm not even sure what I need to ask. I thought I'd try to ...
4
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1answer
97 views

Do the Prime Number Theorem and/or Riemann Hypothesis predict a limit on the accuracy of this formula for $\gamma$?

This question is related to the following formula for Euler's constant $\gamma$ where $A$ is Glaisher's constant. (1) $\quad\gamma=12\,\log(A)-\frac{\pi^2}{6}\sum\limits_{n=1}^N\frac{\mu(n)}{n^2}\,\...
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1answer
69 views

Field with one element and Chaitin's Constant

Some quotes from Wikipedia: In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field ...
3
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1answer
93 views

Statement Equivalent to the Riemann Hypothesis

I am told that the Riemann Hypothesis is equivalent to the condition: $\psi(x) = x + O(x^{1+o(1)})$, and asked to prove this in the forward direction. (Here $\psi(x)$ is the Chebyshev Function). ...
3
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1answer
66 views

Convergence of Riemann spectrum/Fourier transform of prime powers

Prime Numbers and the Riemann Hypothesis by Mazur and Stein makes use of an interesting function: $$\hat{\Phi}_{\le C}(\theta)=2\sum_{prime\:powers\:p^n\le C}p^{-n/2}\cdot log(p)\cdot cos(n\cdot log(p)...
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2answers
218 views

Could this problem be similar to the Riemann Hypothesis?

I've found the below equivalence. For $a,b,x\in\mathbb{C}$, provided that there are no singularities on the right-hand side: \begin{multline}\sum _{k=2}^{\infty}\sum _{j=1}^{\infty}\frac{x^k}{(a ...
6
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2answers
78 views

Can a mathematical proof always be objectively determined as correct or incorrect?

Fields medalist Michael Atiyah claimed a simple proof of the Riemann hypothesis, but many mathematicians rejected his proof. Am I right in saying that Atiyah's proof is either objectively correct (...
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1answer
53 views

Riemann-zeta Function Evaluated at $\zeta(0)$ [duplicate]

WolframAlpha says that $\zeta(0) = - \frac{1}{2}$ but I can't seem to get that result. I found that for $\Re(s) < 1 $, \begin{equation}\label{1} \zeta(s) = 2^s \pi^{s-1}\sin\Bigl(\frac{s\pi}{2}\...
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0answers
71 views

The Meaning of R's non-Trivial Zeros

(I have read through the various similar questions on SE listed by the system but not found an answer that helps). Is there an intuitive explanation for why the Riemann zeta Function (rather than ...
3
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2answers
122 views

The Riemann Hypothesis [duplicate]

(EDIT: I've marked this question as answered in order that I can go away and come up with a better one. Thanks to everybody for the helpful answers.) Is it possible to describe the RH in language ...
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0answers
113 views

Can the Riemann Explicit Formula be used to find prime numbers?

It is well known that there is a strong link between the Riemann Hypothesis and the distribution of primes. The prime number theorem gives the number of primes less than or equal to a given $N$ as: ...
3
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2answers
158 views

A series whose convergence is equivalent to the Riemann hypothesis

It was claimed here that the convergence of the series$$\sum_{n=2}^\infty \frac{\Lambda(n)-1}{n^{1/2}\log^3 n}\tag1$$(where $\Lambda$ is the Von Mangoldt function) is equivalent to the Riemann ...
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0answers
81 views

The Riemann Hypothesis follows from the Polya-Vinogradov inequality?

The Mertens function $M(x)$ is defined as $\Big|\sum_{n\leq x} \mu(n)\Big|$, where $\mu$ denotes the Mobius function. If $\chi$ is a primitive character modulo $q$, the Polya-Vinogradov inequality ...
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1answer
64 views

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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0answers
38 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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45 views

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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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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48 views

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 ...
3
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1answer
480 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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152 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
96 views

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
910 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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69 views

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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0answers
74 views

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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0answers
87 views

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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0answers
31 views

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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177 views

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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0answers
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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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108 views

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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29 views

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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0answers
41 views

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
56 views

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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1answer
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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
85 views

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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2answers
114 views

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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0answers
73 views

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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0answers
156 views

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
238 views

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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0answers
51 views

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
158 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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0answers
41 views

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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1answer
45 views

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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0answers
42 views

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.