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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37 views

Questions on a formula for the Mertens function

The Mertens function $M(x)$ is defined as follows. (1) $\quad M(x)=\sum\limits_{n=1}^x\mu(n)$ I've noticed the Merten's function can also be evaluated as follows which is related to OEIS entry ...
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What is the size of the kth Superabundant Number?

Utilizing the data generated by T.D. Noe for the first $10^6$ terms of A004394, a plot of $k$ against $log_{10}(S(k))$ where S(K) is the $k^{th}$ Superabundant Number shows a strong linear ...
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Riemann Hypothesis and Complex modular exponentiation [on hold]

I have always liked this graph. It looks like modular arithmetic in the 12-hour clock. Is there any relationship of congruence of complex exponents in the critical line? to know the behavior of this ...
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What approach provides the largest known verification of the Riemann Hypothesis?

The Riemann Hypothesis is true if and only if Robin's Inequality is true for all n>5040. It has also been shown by Akbary and Friggstad that the smallest counterexample greater than 5040, if it exists,...
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zeta of imaginary numbers

So the riemman hypothesis was dealing with the real part of the s be 1/2 and I was wondering if it is possible to take zeta of imaginary numbers and if it is, how would you proceed to do so?
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Questions on Riemann Hypothesis prediction and explicit formula for $\hat{\Phi}(x)=\sum\limits_{n\le x}\frac{\phi(n)}{n}$

I've primarily been able to find information on the the summatory Euler totient function $\Phi(x)$ defined in (1) below, but I'm more interested in the $\hat{\Phi}(x)$ defined in (2) below. (1) $\...
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43 views

Is there a formula that connects $\zeta(-s)$ and $\zeta(s-1)$?

I have been learning quite a lot of interesting things about the Riemann zeta function, and I have read that apart from the $\Re(s)>1$ region of the complex plane, the Riemann zeta function at $0&...
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1answer
67 views

The connection between the Riemann hypothesis and the harmonic series

I have heard most people say that the harmonic series, given by $\displaystyle\sum_{n=1}^{\infty}{n^{-1}}$, is central or somehow related to a proof of the Riemann hypothesis. I have been trying to ...
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414 views

A Special Observation on Prime Numbers and $\pi (n)$

$\eth(n)$ is a little algorithm I made, which may appear to be quite complex, so I will start with an example middle of the post. Questions are at the end of the post. Definition Let $...
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How to get to formula for derivative of zeta/zeta

I reading a book for a Seminar, where I have to present some parts out of it. In the book the other says the formula below can be proved by partial integration, but it is not clear for me, hwo that ...
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The Riemann zeta function for complex conjugates

If $\zeta(x)=a+ib$ and $\zeta(y)=a-ib$, is there a single equation that relates $\zeta(x)$ to $\zeta(y)$?
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Formula for the reciprocal of the Hurwitz zeta function

The following equation is for the reciprocal of the zeta function at $Re(s) > 1$: $$ \frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}} = s \int_{1}^{\infty} \frac{M(x)}{x^{1+s}}dx, \ ...
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Some inequalities similar to Lagarias inequality

I have found some inequalities which seem to be equivalent to Lagarias inequality. It would be very nice if someone takes the time to proofread this in detail and give constructive feedback. Thanks ...
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113 views

RH & explicit formula for the number of primes $ \le x$

Does the RH have to be true in order for Riemann's explicit formula for the number of primes <= x to hold? The formula is (copied from wikipedia:https://en.wikipedia.org/wiki/Explicit_formulae_(L-...
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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 ...
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1answer
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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 ?
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1answer
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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\...
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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 $H_G := \sum_{g \in G} \frac{1}{|g|+1}$, ...
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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 ...
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1answer
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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 ...
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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 ...
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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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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 ...
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1answer
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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). ...
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1answer
96 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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1answer
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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 ...
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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
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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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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 ...
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2answers
149 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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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: ...
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2answers
201 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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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
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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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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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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
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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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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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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 ...
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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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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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131 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 ...