Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
48 views

Why the Riemann Zeta function has nontrivial zeros

I know that the Riemann Zeta function has an infinite number of zeros on the critical line $\sigma = 1/2$; that it is possible to determine how many zeros the Riemann Zeta function has on any ...
1
vote
1answer
57 views

Real values of $\frac{\zeta(2 s)}{\zeta(s)}$

If $\frac{\zeta(2 s)}{\zeta(s)}$ is a real number, then must $s$ be real ?
2
votes
1answer
94 views

Are these formulas for the Riemann zeta function $\zeta(s)$ globally convergent?

This question assumes the following definitions. (1) $\quad S(x)=x-\left(\frac{1}{2}-\frac{1}{\pi}\sum\limits_{k=1}^f\frac{\sin(2\,\pi\,k\,x)}{k}\right),\quad f\to\infty$ (2) $\quad S'(x)=1+2\sum\...
1
vote
0answers
77 views

A group theoretic interpretation of Lagarias inequality

Let $G$ be a finite group, $S \subset G$ a generating set. Set $\sigma(G):=\sum_{U \subset G} |U| $, where the sum runs over all subgroups $U$ of $G$. Set $H_G := \sum_{g \in G} \frac{1}{|g|+1}$, ...
1
vote
0answers
34 views

Do the properties defining the Selberg class imply the distribution of real parts of non trivial zeros of an L-function is strongly unimodal?

Selberg defined what is now known as the Selberg class as a class of L-functions fulfilling for essential properties, which are analyticity, Euler product, functional equation and Ramanujan-Patersson ...
1
vote
1answer
70 views

The equation $\zeta(q)=0$ for $q$ a quaternion

I know there have been several attempts to define a theory of functions of a quaternionic variable. I would like to know if a coherent and satisfying definition of the "Riemann" zeta function exists ...
0
votes
0answers
41 views

Root objects and the simplest possible analytic continuation of the Riemann zeta function.

The equation I am trying to solve is: $$\lim\limits_{k \rightarrow 3} \left( \sum\limits_{n=1}^{n=k} \frac{1}{n^s}+ \frac{1}{k^{s - 1} \cdot (s - 1)}\right)=0 \tag{1}$$ The simplest possible ...
4
votes
1answer
149 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}\,\...
-2
votes
1answer
75 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
votes
1answer
112 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
votes
1answer
86 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)...
4
votes
2answers
264 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
votes
2answers
83 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 (...
0
votes
1answer
57 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}\...
2
votes
0answers
86 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
votes
2answers
138 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 ...
5
votes
0answers
124 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
votes
2answers
177 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 ...
1
vote
0answers
84 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 ...
0
votes
1answer
89 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....
0
votes
0answers
40 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^{-...
1
vote
0answers
47 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 ...
1
vote
0answers
100 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 ...
1
vote
0answers
54 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
votes
1answer
487 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 ...
0
votes
0answers
160 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 $|t|=|\Im(s)|\geq 165$ ...
8
votes
1answer
170 views

Closed form expression or asymptotic expansion for (periodic) generalized harmonic numbers?

In contrast with the series $\sum_{k=1}^n k$ and $\sum_{k=1}^n1$, there does not (as far as I know) exist a pure closed form expression (or a nice asymptotic expansion other than the Euler-Maclaurin ...
0
votes
1answer
105 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)=...
5
votes
1answer
919 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\...
0
votes
0answers
72 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 $$\displaystyle\int_1^{+\infty} \lfloor x\rfloor x^{-2-\rho} dx =\int_1^{+\infty} \lfloor x\rfloor \{ x \}x^{-2-\rho} dx $$ ...
1
vote
0answers
85 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 ...
1
vote
0answers
90 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 ...
1
vote
0answers
33 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 ...
1
vote
0answers
184 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 ...
0
votes
0answers
39 views

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 ...
3
votes
0answers
121 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 ...
0
votes
0answers
29 views

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 ...
0
votes
0answers
18 views

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 ...
0
votes
0answers
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 ...
0
votes
0answers
44 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) $\...
3
votes
0answers
137 views

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. ...
0
votes
1answer
66 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 ...
2
votes
0answers
93 views

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) $\quad f(x)=\sum\limits_{n=1}^{x}A143519(n)$ ...
42
votes
1answer
10k views

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?
0
votes
0answers
241 views

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 ...
8
votes
1answer
1k views

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 ...
0
votes
1answer
90 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 ...
2
votes
0answers
101 views

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 ...
0
votes
2answers
115 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 ...
2
votes
0answers
75 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 $...