# Questions tagged [riemann-hypothesis]

Questions about the famous conjecture from Riemann saying that the non-trivial zeroes of the Riemann Zeta function all lie on the so-called critical line $\Re(s)=\dfrac{1}{2}$, its various generalizations and the different approaches towards its solution.

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### Chebyshev's bias-conjecture and the Riemann Hypothesis

Chebyshev's bias-conjecture that says "there are more primes of the form 4k + 3 than of the form 4k + 1" and the Riemann Hypothesis are equivalent? That means, one implies the other (if and only if)?

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**1**answer

168 views

### Is there an analogue of the Balazard-Saias-Yor criterion for the Riemann Hypothesis for finite fields?

The Balazard-Saias-Yor criterion for the Riemann Hypothesis states that the latter is equivalent to the statement that
$$\int_{\Re(s)=1/2} \frac{\log|\zeta(s)|}{|s^2|}|ds|=0$$ where $\zeta$ denotes ...

**1**

vote

**1**answer

544 views

### An integral involving the argument of the Gamma function and the Riemann Hypothesis

Evaluate $$I=\int_{0}^{\infty} \frac{t\arg
\Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$
where $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$
Note that $I$ converges ...

**2**

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**0**answers

174 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 ...

**3**

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**0**answers

193 views

### Is there an analogue for the Balazard et al criterion for the Generalised Riemann Hypothesis?

A nice result of Balazard et al says the Riemann Hypothesis is equivalent to the statement that
$$\int_{-\infty}^{\infty} \frac{\log|\zeta(1/2 + it)|}{\frac{1}{4}+t^2} \mathrm{d}t=0$$ where $\zeta$ ...

**14**

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**1**answer

371 views

### What is the automorphic interpretation of the Weil conjectures over finite fields

I am very much a beginner in the theory of automorphic forms and I might (will?) make mistakes in what follows. Please correct me.
A loose interpretation of the Langland's philosophy is that to any ...

**2**

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188 views

### Does the Riemann Xi function possess the universality property?

Here is the question.
Does the Riemann Xi function possess the universality property, or something similar to Voronin 's universality property?
This is the reason why the answer to this question ...

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**1**answer

408 views

### Where can i find LeClair's 2007/8 papers on the Riemann Hypothesis?

I'm reading Schumayer's excellent expository article, ''Physics of the Riemann Hypothesis'', and on pp. 21-22 in the article, Schumayer refers to certain two papers of LeClair, published in 2007 and ...

**4**

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261 views

### $\sum_{n=1}^\infty \Lambda(n) e^{-nz}$ and $L(s,\chi)$

Let $$W(z)=\sum_{n=1}^\infty \frac{\Lambda(n)}{n^{1/2}} e^{-nz}, \qquad\Re(z) > 0$$ For $\frac{y}{2\pi}=\frac{a}{q} \in \mathbb{Q}$, as $x \to 0^+$ we have $$W(x-iy) -{\scriptstyle \underset{(n,q) &...

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725 views

### Euler's totient function and Riemann hypothesis

I am looking for an upper-bound of the Euler's totient function $\varphi$ which would be equivalent to the Riemann hypothesis (RH). There is the following Nicolas' criterion about primorial numbers $...

**3**

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**1**answer

164 views

### Almost-Primes in Short Intervals

Let $S$ be the set of integers which are a product of $k$ distinct primes, $k$ a fixed positive integer (the condition that the primes are distinct is not crucial). Landau used the Prime Number ...

**98**

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**1**answer

29k views

### What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?

In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the ...

**15**

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**2**answers

528 views

### Analogues of the Riemann zeta function that are more computationally tractable?

Many years ago, I was surprised to learn that Andrew Odlyzko does not consider the existing computational evidence for the Riemann hypothesis to be overwhelming. As I understand it, one reason is as ...

**47**

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2k views

### Motivated account of the prime number theorem and related topics

Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often ...

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230 views

### Order of magnitude of extremely abundant numbers and RH

I have always been intrigued by the fact that Riemann's hypothesis is equivalent to the assertion (you can find the scanned paper here) that the inequality $$\frac{\sigma(n)}n<e^\gamma \log\log n \...

**28**

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**1**answer

1k views

### On a quantum Riemann Hypothesis

Robin's theorem (1984) states that
$$ \sigma(n) < e^\gamma n \log \log n$$
for all $n > 5040$ if and only if the Riemann hypothesis is true.
Recall that $γ$ is the Euler–Mascheroni ...

**10**

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**1**answer

491 views

### How many zeta zeros are needed to accurately calculate five digits for π(1000000), where π(x) is the prime counting function?

John Derbyshire in his book PRIME OBSESSION says on page 343:
"I’ll round off with a complete calculation of $\pi(1,000,000)$, the
number of primes up to one million, using Riemann’s formula -- ...

**4**

votes

**1**answer

254 views

### Confusion about Montgomery's Pair Correlation Conjecture

This question will be based roughly on the Bourgade Keating review on Zeta function and eigenvalue asymptotics (BK): https://link.springer.com/chapter/10.1007/978-3-0348-0697-8_4
To set up the ...

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**0**answers

488 views

### What is your favourite wrong proof of RH? [closed]

Some of the users here receive claimed proofs of the Riemann hypotheses on a regular bases. As fas as we know all of them have been wrong. But sometimes failure is also interesting.
So for all cases ...

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**1**answer

1k views

### Does $ M(x)=O(\sqrt{x}) $ if and only if the De Bruijn-Newman constant is negative?

The Riemann hypothesis is equivalent to the assertion that the De Bruijn-Newman constant $ \Lambda $ , as defined in https://www.sciencedirect.com/science/article/pii/S0001870809001133/pdf?md5=...

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**0**answers

131 views

### Permutation of the natural numbers related to the Mobius function and relation to the Riemann Hypothesis

Let $f:\mathbb{N}\rightarrow \mathbb{N}$ such that:
1) $f$ is bijective.
2) $f(n)=m \Rightarrow f(m)=n$
3) $f(n)=m \Rightarrow \mu(n)+\mu(m)=0$
We also require $f(n)$ to be as close to $n$ as ...

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554 views

### GRH and the rank of elliptic curves

I have been using the Magma calculator recently, and while calculating ranks of elliptic curves with very big coefficients, there is a possibility to assume GRH is true, which signaficantly speeds up ...

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**1**answer

598 views

### Has there been further work on Bender-Brody-Müller approach to RH?

Earlier this year (April 4 2017), a seemingly tantalizing approach of the Riemann Hypothesis based on ideas dating back to Hilbert and Polya by Bender, Brody and Müller was made publicly available. I ...

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**1**answer

1k views

### Is this equivalent to RH - Riemann hypothesis?

$$\pi = 3\prod_{\zeta(1/2+it) = 0}\frac{9+4t^2}{1+4t^2}\iff\text{RH is true}.$$

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**1**answer

758 views

### How does Riemann hypothesis implies estimates?

In Iwaniec, Luo and Sarnak article (precisely (4.23)), it is said that GRH for $L(s, \mathrm{sym}^2(f))$, for a holomorphic cusp newform $f$ of level $N$ and weight $k$, implies
$$\sum_{p \nmid N} \...

**3**

votes

**3**answers

322 views

### zeros of a complex function defined by integers

This is a crosspost from Math.SE.
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 ...

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**3**answers

7k views

### Fake integers for which the Riemann hypothesis fails?

This question is partly inspired by David Stork's recent question about the enigmatic complexity of number theory. Are there algebraic systems which are similar enough to the integers that one can ...

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**1**answer

297 views

### Is any $n>60$ known to have a divisor sum greater than $e^{H_n}\log({H_n})$, where $H_n$ is the nth harmonic number?

The Riemann hypothesis is equivalent to
$\forall n\geqslant2,\: \sigma(n)<H_n+e^{H_n}\log{H_n}$,
where $\sigma(n)$ is the divisor sum of $n$ and $H_n$ is the nth harmonic number.
For large $n$, $...

**2**

votes

**1**answer

404 views

### On Robin's criterion for the Riemann Hypothesis

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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387 views

### Riemann hypothesis for the Hecke operators and modular forms

Let $f(z) = \sum_{n=1}^\infty a(n) e^{2i \pi nz}$ be an eigenform of $S_k(\Gamma_0(N))$. Since the Hecke operator acts by $T_p f = a_p f$ the Riemann hypothesis for $f$'s L-function is
$$ \!\!\! \!\...

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**1**answer

538 views

### Seek a reference for Theorem 1.2 on p. 6 of the Riemann Hypothesis sourcebook of Borwein et. al

The book "The Riemann Hypothesis, A Resource for the Afficionado and Virtuoso Alike", Borwein, Choi, Rooney, Weirathmueller, Eds., states on its page six the following theorem (Theorem 1.2):
The ...

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**1**answer

688 views

### The connection between the Weil conjectures and Ramanujan's conjecture

I'm writing an essay about Ramanujan's conjecture and have some questions:
1 How is Ramanujan's conjecture connected with the Weil conjectures?
2 How could Ramanujan's conjecture be assumed true or ...

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votes

**1**answer

517 views

### Riemann Hypothesis and Euler product

It is conjecture that under certain conditions a L-function satisfies RH.
Among these conditions there is the necessity for the L-function to have an Euler product. (Some L-functions with a functional ...

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**2**answers

433 views

### What is the physical interpretation of the Riemann Hypothesis? [closed]

Some propositions in math can be modeled as a physical system. Has anyone done this for RH?

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1k views

### PT symmetry and the Riemann Hypothesis

Recently there have been articles in Quanta, in Science Alert, and at phys.org among others, on possible recent progress toward the Hilbert-Polya conjecture, which implies the Riemann Hypothesis. The ...

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315 views

### Numerical Evidence for Grand Riemann Hypothesis?

Let $L(s)$ be an $L$-function coming from Hecke characters or automorphic forms (e.g. modular form on GL(2), Maass form on GL(2), and higher-rank analogues).
Is there any numerical evidence for ...

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**2**answers

734 views

### Largest known zero of the Riemann Zeta function

Numerical calculations on the zeroes of the Riemann Zeta function have reached a very high degree of refinement and sophistication and I think that the first $10^{20}$ (with positive imaginary part) ...

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153 views

### R.H. equivalent statement condition

Is the inequality $\prod \limits_{p \leq \sqrt{x}} (1+\frac{1}{p^2-1}) \prod \limits_{p \leq x} (1+\frac{1}{p}) \leq e^\gamma \ln(\theta(\sqrt{x})+\theta(x))$ where $\theta(x)$ is the Chebyshev's ...

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**1**answer

261 views

### Numbers related to the Riemann hypothesis

Are there numbers $k > 1$ and $c > 1$ such that:
1 ) $\theta(c) \geq c \left( 1-\frac{1}{5 \ln^2(c)} \right) $
2 ) $\frac{c}{1+\frac{1}{\ln^4(c)}} \leq p(\pi(c))$ where $p(n)$ is the $n$-th ...

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85 views

### Explicit formula for k-central numbers

Given a positive integer $ n $ and assuming Goldbach's conjecture, let $r_{0}(n)$ denote the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are primes. Let $k_{0}(n)$ denote 'the ...

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**1**answer

323 views

### Related Forms for the Riemann Hypothesis over Function Fields

There are several formulations and consequences of the Riemann Hypothesis over Function Fields (RH, from now on). I am interested in the logical implications between those, and in proofs\references ...

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**0**answers

476 views

### Result About Zeta Function Worth Publication? [closed]

Over the past two years at my high school a friend and myself have been working on a paper about the Riemann Hypothesis.
In this paper we derive several definitions of the zeta function from the ...

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**1**answer

381 views

### Is Selberg's eigenvalue conjecture related to RH?

I took a quick glance on a survey paper about superzeta functions where one considers a pair $\rho\leftrightarrow 1-\rho$ of non trivial zeroes of the Riemann zeta function. The assumption of RH, i.e $...

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**1**answer

329 views

### Prime quadratic non-residue

NC Ankeny showed assuming Riemann Hypothesis the least quadratic non residue( let it be '$r$') modulo some prime $p$ to be $O(\log^2 p)$. It is easy to see that $r$ is a prime.
I have following ...

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**2**answers

700 views

### Statements going against the grain of Riemann Hypothesis (R.H.) [closed]

Let $M(N) := \sum_{n=1}^N \mu(n)$
It is known that bounding $M(N)$ by $N^{1/2+\epsilon}$ implies R.H.
A bound of $M(N)$ by $K\sqrt N$ for say $ K\ge2 $ is also sufficient. $K = 1$ is excluded as ...

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**1**answer

252 views

### Lexicographic distribution of irreducible polynomials

Let $A = {\mathbb F}_2[X]$, though the following can be adapted to $p \neq 2$ too. Order the elements of $A$ lexicographically. Equivalently, take a polynomial such as $P = X^4 + X + 1$, write its ...

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240 views

### On a sequence of L-functions having same zeros in critical strip and GRH

I had an idea on GRH involving a sequence of L-functions having same zeros, then at one step I need a bound on these function and I wonder if this bound is in fact not as hard as GRH itself ?
Let's ...

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438 views

### Riemann hypothesis in Zilber's field

Question. What is known about the situation (truth or falsity) of Riemann hypothesis in the Zilber's field?

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**1**answer

2k views

### Can one expect the existence of a relevant approach for a proof of the Riemann hypothesis using Mochizuki's theory? [closed]

Next month at Oxford university, there will have the first workshop outside Asia on the Inter-Universal Teichmuller theory of Shinichi Mochizuki: http://www.claymath.org/events/iut-theory-shinichi-...

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**2**answers

416 views

### Does theta(n)<n for all n imply the Riemann Hypothesis and/or vice versa?

I know that better and better bounds of the Chebyshev Theta and Psi functions are implied by knowing that the first (insert large number here) zeta zeroes lie on the Critical Line. These bounds, ...