# 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.

156
questions

**7**

votes

**2**answers

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**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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

**1**answer

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

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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**

vote

**0**answers

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

**1**answer

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

**3**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**5**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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=...