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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7
votes
2answers
557 views

What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?

From Terry Tao's post here there is the statement: "Conversely, if one can somehow establish a bound of the form $$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$ ...
0
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0answers
76 views

On the asymptotics of the Chebyshev psi function

Denote by $\psi(x)$ the Chebyshev psi function over prime powers. Assuming the RH, it can be shown via the Riemann explicit formula that $$\psi(x)-x \ll √x |\sum_{|\gamma| < T} \frac{1}{\rho} | +...
5
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0answers
168 views

Explicit Formula for $n$th prime in terms of Riemann zeros?

We all know there exists a explicit Formula for prime counting function in terms of Riemann zeros. I'm wondering if similar formula exists for $n$th prime in terms of Riemann zeros?
5
votes
1answer
226 views

Proving a specific case of Robin's Inequality

Edit: It turns out that this is equivalent to the RH which gives the idea that this might a a little difficult to show. As such we could consider an even simpler case in which the number $n$ is ...
0
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0answers
69 views

Equivalent or corollary to Nicolas' theorem

Nicolas proved that R.H. is true iff $e^{\gamma} \log(\theta(n)) < \prod \limits_{p \leq n} \frac{p}{p-1}$ for all integer $n \geq 2$ One can derive easily from the above that $\log \log (\theta(n)...
2
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0answers
174 views

Riesz equivalent of Riemann hypothesis and Hadamard product

First define Hadamard product : Conider two power series is definded as follows : $$ f(x) = \sum_{n=0}^\infty f_n x^n $$ $$ g(x) =\sum_{n=0}^\infty g_n x^n $$ Then , Hadamard product of $f$ and $g$:...
0
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1answer
454 views

On provability of false statements in constructive mathematics [closed]

Lagarias "elementary" reformulation of Robin's theorem is that $$\mathrm{RH}\iff\sigma(n)\leq H_n+e^{H_n}\log(H_n)$$ holds for every $n\geq 1$, where $\sigma(n)$ is the sum of divisors function and $...
4
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0answers
513 views

Has any professional mathematician ever attempted to solve the Riemann hypothesis using only number theory? [closed]

I have often heard people saying that ''all attempts at solving the Riemann hypothesis using number theory have failed.'' But in the literature, i cannot find any failed ''purely number-theoretic'' ...
6
votes
1answer
528 views

Attempt at applying linear programming to the partial sums of the Möbius inverse of the Harmonic numbers

Let $a(n)$ be the Dirichlet inverse of the Euler totient function: $$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$ and let the matrix $T(n,k)$ be: $$T(n,k)=a(\gcd(n,k)) \tag{2}$$ It has been ...
3
votes
0answers
132 views

Largest observed value of $S(t)$

Let $S(t)$ be the deviation of the number of zeros of the Riemann zeta function up to height $t$ from the expectation. What is the largest observed value of $S(t)$ today? Here is a quote from a ...
10
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406 views

Bounding $1/\zeta(s)$ given RH

Let $T\geq 0$. Assume RH(T+100), that is, assume that all non-trivial zeros $\rho$ of the Riemann zeta function with $|\Im(\rho)|\leq T+100$ satisfy $\Re(\rho)=1/2$. Can we then give a good upper ...
10
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0answers
332 views

On Riesz criteria for Riemann hypothesis:

While Reading the book "Equivalents of Riemann hypothesis" by Kevin Broughan I came across the Riesz criteria for Riemann hypothesis . Riesz defined a function : $R(x) = \sum_{n=1}^\infty \frac {(-1)...
2
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0answers
404 views

On a Possible equivalent of Riemann hypothesis

I've read in a Bombieri's paper on official problem statement of Riemann hypothesis for Clay Math institute's millennium problems, a statement and what I understood of it is the following : The ...
19
votes
1answer
978 views

Is the Hilbert–Pólya intuition vindicated in the function field case?

The Hilbert–Pólya conjecture is the name given to the idea that the "reason" or "explanation" for the collinearity of the non-trivial zeros of the Riemann zeta function $\zeta(s)$ is that they are the ...
2
votes
1answer
170 views

Interpretation of an equivalence to the Riemann hypothesis due to de Reyna and Toulisse in the spirit of a formula from an article

In [1] the authors present an equivalence to the Riemann hypothesis that is the Theorem 6.2. On the other hand I know a statement from [2], in English this is the article Andrew Granville and Greg ...
2
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0answers
124 views

On the connection between $\pi(x)-Li(x)$ and $\theta(x)-x$

Let $\pi(x)$ be the number of primes $p$ not exceeding $x, \theta(x) = \sum_{p\leq x} \log p$ and $Li(x)$ be the logarithmic integral. Is it true that $$\pi(x)-Li(x) = \theta(x) - x + O(x^{1/2}\log^{...
2
votes
1answer
417 views

Books on complex analysis for self learning that includes the Riemann zeta function?

I am searching for an introductory book in the field of complex analysis for self learning, that would contain the following: Analytic number theory : the connection between complex analysis and ...
6
votes
1answer
605 views

Riemann Hypothesis, Primes and Groups

Let $G$ be a finite group $S\subset G$ a generating set $|g|=$ word length with respect to $S$. Set $$ \sigma(G) = \sum_{H \le G} [G:H]$$ Let $\rho$ be the regular representation and set $A_G := \...
2
votes
1answer
372 views

Conditional bound on RH for $\Re\left(\sum_{p\leq\sqrt{x}}\frac{(1/2)}{p^{1+2it}}\right)$

I would like to prove that Assume RH. Let $T$ large, $2\leq x \leq T^2$ and $T\leq t \leq 2T$, then $$ \log|\zeta(1/2+it))|\leq \Re\left(\sum_{p\leq x}\frac{1}{p^{1/2+1/\log x+it}}\frac{\log(x/p)}...
30
votes
2answers
2k views

What is the most direct proof of the Riemann hypothesis for Dirichlet L-functions over function fields?

Let $\mathbb{F}$ be a finite field of order $q$, let $m$ be an irreducible polynomial in the ring $\mathbb{F}[T]$, and let $\chi$ be a Dirichlet character modulo $m$. Define the function field ...
2
votes
0answers
124 views

A question about Lagarias inequality

Let $|g|=\min(g,n-g)=$Lee-norm on $\mathbb{Z}/(n)=$ word length on cyclic group $C_n$ with respect to generating set $S=\{\pm 1\}$. Let $H^{L}(C_n) := \sum_{g \in C_n} \frac{1}{|g|+1}$, which as one ...
10
votes
1answer
544 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 $H_G := \sum_{g \in G} \frac{1}{|g|+1}$, ...
7
votes
1answer
242 views

Subspaces of $L^2(0,1)$ dense on every truncation $L^2(c,1)$

It may be better to move this to a separate question. Let me call a linear subspace $V \subset L^2(0,1)$ to be tame if, for every linear subspace $W \subset V$, either $W$ is dense in $L^2(0,1)$, or ...
6
votes
1answer
367 views

Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$?

Added. My question in the title was solved (in the negative) by Nik Weaver (in the answer below) and Mateusz Kwaśnicki (in the comments). In both solutions, the reason is that the $L^2$ density fails ...
1
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0answers
140 views

Can Riemann's explicit formula be generalized to semi-primes?

Following Isometry group of an integer I wonder if one can define a "mock zeta function" $\zeta_{V}$ (where $V:=(\mathbb{Z}/2\mathbb{Z})^{2}$ stands for "Vierergruppe", the German word for the Klein ...
2
votes
1answer
194 views

Upper bounds for $|\theta(x)-x|$ assuming Riemann Hypothesis

What are the best currently known upper bounds for $|\theta(x)-x|$ assuming the Riemann Hypothesis, where $\theta(x)$ is the Chebyshev theta, and can someone provide the reference for this (not ...
31
votes
3answers
5k views

Why is so much work done on numerical verification of the Riemann Hypothesis?

I have noticed that there is a huge amount of work which has been done on numerically verifying the Riemann hypothesis for larger and larger non-trivial zeroes. I don't mean to ask a stupid question, ...
9
votes
0answers
294 views

Additive and multiplicative convolution deeply related in modular forms

From the fact spaces of modular forms are finite dimensional, from the decomposition in Hecke eigenforms, and from the duplication formula for $\Gamma(s)$ there are a lot of identities mixing additive ...
1
vote
0answers
208 views

Riemann Explicit Formula

I am writing my senior thesis on Montgomery's pair correlation conjecture, and in his first lemma, he uses the following explicit formula: $$\sum_{n \leq x} \Lambda(n) n^{-s} = -\frac{\zeta'}{\zeta}(s)...
10
votes
1answer
817 views

What really is the link between quantum gravity and the Riemann Hypothesis that was speculated by Connes and Marcolli?

In their book, ''Noncommutative Geometry, Quantum Fields and Motives,'' Alain Connes and Matilde Marcolli begin their preface by saying: The unifying theme, which the reader will encounter in ...
1
vote
2answers
364 views

Questions concerning the Fourier analysis of $ nx\ \%\ m$

Let $x\ \%\ m$ be the residue of $x$ modulo $m$, i.e. $$x \equiv x\ \%\ m\pmod{m}$$ The plots of the functions $f_{nm}(x) = nx\ \%\ m$ exhibit characteristic patterns, especially periods of length $...
4
votes
2answers
878 views

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)?
2
votes
1answer
215 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
1answer
623 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
votes
0answers
198 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 ...
15
votes
1answer
441 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
votes
0answers
265 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 ...
1
vote
1answer
423 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 ...
5
votes
0answers
361 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) &...
10
votes
0answers
933 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
votes
1answer
179 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 ...
105
votes
1answer
30k 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
votes
2answers
609 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 ...
46
votes
5answers
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 ...
8
votes
0answers
239 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 \...
29
votes
1answer
2k 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 ...
12
votes
2answers
672 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
1answer
457 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 ...
7
votes
0answers
606 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 ...
5
votes
1answer
2k 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=...