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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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Prime differences and zero multiplicity

Concerning gaps between consecutive primes, Paul Erdős conjectured that: $$\sum_{p_n < x} (p_n -p_{n-1})^2 = O(x \log x)$$ Let's call this hypothesis EH. Assuming the Riemann hypothesis (RH), ...
Felixson's user avatar
  • 232
-2 votes
2 answers
303 views

Bounds for analytic circles

It is known that for certain particular entire functions $f(s)$ of first order, in the circle $|s| = p$, if $\epsilon$ is a positive number as small as desired, the following bound holds: $$|f(s)| = O(...
Bo Jonsson's user avatar
0 votes
0 answers
112 views

Convergence of a series related to counting distinct prime factors

I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at ...
Zachary Hoelscher's user avatar
1 vote
0 answers
140 views

Behavior of Dirichlet L-functions at the edge of the critical strip

Given a Dirichlet L-function $L(\chi, s)$ of a primitive character $\chi$, what is the asymptotic behavior of $L(\chi, 1+it)$ for real $t$? I am looking for as many answers for the same question. This ...
edward cornfoot's user avatar
0 votes
0 answers
393 views

Spiegel Vermutung: no Siegel zeros iff GRH is equivalent to Goldbach's conjecture

I apologize for using German language in the title, but this question came to my mind after watching the French movie "le théorème de Marguerite" in which the protagonist gets an insight ...
Sylvain JULIEN's user avatar
2 votes
0 answers
136 views

Translation of an article of Littlewood

I want to read the English translation of an article of Littlwood titled "Quelques conséquences de l'hypothese que la fonction $ζ (s)$ de Riemann n'a pas de zéros dans le demi-plan $ℜs> 1/ 2$.&...
Eloon_Mask_P's user avatar
4 votes
1 answer
568 views

Is $\sum_{n\leq x}{z^{\Omega(n)}} = O(x^{\frac12 + \varepsilon})$ equivalent to the Riemann hypothesis for all roots of unity $z\neq1$?

$\Omega(n)$ is the number of prime divisors of $n$, counted with multiplicity. For $z=-1$, $z^{\Omega(n)} = \lambda(n)$ is the Liouville function, and it's known that $\sum_{n\leq x}\lambda(n) = O(n^{\...
Command Master's user avatar
1 vote
0 answers
239 views

Riemann hypothesis and zero of Mellin transform of eigenfunction of Schrödinger equation

The paper [1] shows that the eigenfunction of the Schrödinger equation $$ \left(x^2-\frac{1}{4\pi}\frac{d^2}{dx^2}\right)f_n=\frac{2n+1}{2\pi}f_n $$ satisfies the same functional equation as the ...
George's user avatar
  • 221
0 votes
0 answers
111 views

Question on the inverse Mellin transform $p(x)=\mathcal{M}_s^{-1}\left[-\xi(s)\,\frac{\zeta'(s)}{s\,\zeta(s)^2}\right]\left(\frac{1}{x}\right)$

Consider the function $$p(x)=\underset{K\to\infty}{\text{lim}}\left(\sum\limits_{k=1}^K \Lambda(k) \left(\frac{2 \pi k^2}{x^2}-1\right) e^{-\frac{\pi k^2}{x^2}}\right)\tag{1}$$ where $$P(s)=s\, \...
Steven Clark's user avatar
  • 1,081
2 votes
1 answer
702 views

Does the Riemann hypothesis predict a bound for this prime-counting function?

Does the Riemann hypothesis predict an upper bound for $$\left|f(x)-\left(\operatorname{li}(x)-\frac{x}{\log x} \right)\right|,\quad x\ge 2\tag{1}$$ where $$f(x)=\sum\limits_{n=2}^x \frac{\Lambda(n)}{\...
Steven Clark's user avatar
  • 1,081
3 votes
0 answers
231 views

Is this irregular curve asymptotic to $\log (2) (\log (x)-\log (2))^2-\frac{\log (2)}{2}$, or is the asymptotic something else?

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 $T$ be: $$T(n,k)=a(\gcd(n,k)) \tag{2}$$ which has the property ...
Mats Granvik's user avatar
  • 1,133
1 vote
1 answer
333 views

Hilbert–Pólya conjecture with Grommer inequalities?

The Grommer inequalities are equivalent to RH and formulated on page 20 of Conrey - Riemann's hypothesis: Let $$\Xi(t) := \xi(1/2+it).$$ Then RH is equivalent to : All zeros of $\Xi(t)$ are real. The ...
mathoverflowUser's user avatar
3 votes
1 answer
297 views

Zeros of the derivative of $\xi$

In his paper on Zeros of derivatives of Riemann $\xi$-function on critical line Brian Conrey mention that It can be shown that the Riemann hypothesis implies that all zeros of $\xi (s)$, the ...
Tokita Ohma's user avatar
0 votes
0 answers
232 views

Is it possible to bound Mertens function $M(n)$ from an inclusion-exclusion formulation?

In this post I proposed a formulation of Mertens function $M(n)$ using the inclusion-exclusion principle, as follows: $$M(n)=-\pi\left(n\right)+\left(\sum_{p_{i}<\sqrt{n}}\pi\left(\lfloor\frac{n}{...
Juan Moreno's user avatar
2 votes
0 answers
149 views

Tools to prove lower bounds in analytic number theory

Perron's formula and related methods are used to relate statements such as the Riemann hypothesis to upper bounds of functions occurring in analytic number theory. For example, Perron's formula is ...
EGME's user avatar
  • 1,018
-4 votes
1 answer
444 views

What is the proof for any non trivial zero? [closed]

There are many known nontrivial zeros of the Riemann Zeta function, but I have never seen proof that any of them actually resolve to zero. The trivial zeros make sense because there is a more ...
Not Riemann's user avatar
1 vote
1 answer
262 views

GRH and the Euler product

Let $L(\chi, s)$ be the Dirichlet L-Function of a primitive character $\chi$. I believe, if I’m not mistaken, the convergence of the Euler product of $L(\chi, s)$ in the critical strip is known to be ...
edward cornfoot's user avatar
1 vote
0 answers
3k views

Contribution of Yitang Zhang latest results if correct to correlation conjecture of H. L. Montgomery?

There are some integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. One of them is the integral introduced by Selberg related to estimating the variance of primes in ...
zeraoulia rafik's user avatar
2 votes
0 answers
198 views

An approach to the prime number theorem with Rademacher variables and a recursive formula for the prime pi function?

Consider the bipartite graphs defined here: Why is this bipartite graph a partial cube, if it is? We do random walks on them with equal propability and since the graphs are finite and connected the ...
mathoverflowUser's user avatar
1 vote
1 answer
230 views

Robin's inequality for odd numbers

In this article (Theorem 1.2) there is a proof for Robin's inequality for odd numbers, $\sigma(n)/n< e^{\gamma}\log(\log(n))$ where $\gamma$ is the Euler-Mascheroni constant and $\sigma(n)$ is the ...
Asanovic Tomas's user avatar
10 votes
0 answers
753 views

Example of sequence of graphs which satisfy the Riemann hypothesis or the prime number theorem?

Let us look at the sequence of bipartite graphs $G_n = (V_n, E_n)$ where $V_n = A_n \cup B_n$ defined in this quesiton: Why is this bipartite graph a partial cube, if it is? . The shortest path ...
mathoverflowUser's user avatar
6 votes
2 answers
854 views

Formalisation of the Riemann Hypothesis

Might there be a research team that has formalised the Riemann Hypothesis? So far I have encountered two related questions: Is there a formulation of the Riemann Hypothesis in first-order arithmetic? ...
Aidan Rocke's user avatar
  • 3,827
4 votes
0 answers
191 views

A combinatoric generalization of Zeta

Recall the well known identity $\zeta(s)=\prod_p \frac{1}{1-p^{-s}}=\sum_{n=1}^{\infty} \frac{1}{n^s}, Re(s)>1$ Now take a infinite discrete sequence of real values $r_k$ such that $r_1<r_2<.....
Gui's user avatar
  • 67
0 votes
1 answer
194 views

Spacings of Satake parameters under Ramanujan conjecture

I would like to know if, under Ramanujan conjecture, the following three distributions are known or conjectured to match: the distribution of spacings between Satake parameters of an L-function $F$ ...
Sylvain JULIEN's user avatar
2 votes
1 answer
386 views

'Almost all' zeros of the Dirichlet L function lies 'near' the critical line?

Is there a well known result that states that as $t \to \infty$, 'almost all' zeros of any Dirichlet L function $L(s,\chi)$ lie in the region $R= \{\sigma+i t\mid |\sigma -\frac{1}{2}| \leq \Phi(t) \}$...
Aritro Pathak's user avatar
2 votes
1 answer
721 views

Does asymptotic Goldbach imply GRH?

It seems to me that a proof of $\alpha_{n}=o(n)$ where the quantity $\alpha_{n}$ is defined in About Goldbach's conjecture together with the main result of https://kyushu-u.pure.elsevier.com/en/...
Sylvain JULIEN's user avatar
1 vote
0 answers
153 views

Normal numbers and law of the iterated logarithm

If I remember correctly, for the binary digits of a real number in $[0,1]$, I was told that satisfying the law of the iterated logarithm (LIL) is stronger than being normal. That is, supposedly, some ...
Vincent Granville's user avatar
4 votes
1 answer
295 views

Selberg class definition and Riemann hypothesis

Looking at the Selberg class definition on Wikipedia, under "Comment on definition", there is this paragraph: "The condition that the real part of $\mu_i$ be non-negative is because ...
Bertrand's user avatar
  • 1,121
9 votes
0 answers
414 views

From holes in the image of peculiar functions to new perspective on the Riemann Hypothesis

I am working with the Dirichlet eta function $\eta(z)$, with $z=\sigma+it$, $\sigma > \frac{1}{2}$, and $t>0$. Let us define $$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
Vincent Granville's user avatar
0 votes
0 answers
141 views

Does it make sense to express upper bounds on arithmetic sequences with Dirichlet generating functions?

In order to see what happens when taking the functional equation in this form: $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s) $$ $$\xi(s) = \xi(1 - s)$$ $$\pi^{-s/2}\ \Gamma\left(\...
Mats Granvik's user avatar
  • 1,133
5 votes
2 answers
934 views

A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis

Let $p_n$ be the $n$th prime. Assuming the Riemann hypothesis, Harald Cramér proves that $p_n-p_{n-1}\le C(\sqrt p_n \log p_n)$ for sufficiently large $n$. Is there a value known for the constant $C$ ...
EGME's user avatar
  • 1,018
3 votes
0 answers
4k views

Intuition for the bias of the partial sums of the Liouville function

It's a well known result that the Dirichlet series of the Liouville function $ \lambda(n) $ is given by $$ \sum_{k=1}^{\infty} \frac{\lambda(k)}{k^s} = \frac{\zeta(2s)}{\zeta(s)} $$ If we use Perron's ...
Ege Erdil's user avatar
  • 291
4 votes
2 answers
411 views

Is the function $F(x) = \exp(x) + \exp(\exp(x))x$ a hypertranscendental function?

The function $F(x) = \exp(x) + \exp(\exp(x))x$ plays a role in the formulation of the Lagarias inequality: $$\sigma(n) \le H_n + \exp(H_n) \log(H_n)$$ If we put $x = \log(H_n)$, then this inequality ...
mathoverflowUser's user avatar
4 votes
1 answer
313 views

Double sum over zeros of Riemann zeta-function

In a paper by Saffari and Vaughan there appears a complicated-looking double sum $$\Sigma_1=\sum_{\rho_1}\sum_{\rho_2}\frac{(1+\theta)^{\rho_1}-1}{\rho_1}\cdot \frac{(1+\theta)^{\bar{\rho_2}}-1}{\bar{\...
Daniel Johnston's user avatar
1 vote
0 answers
187 views

Is it possible in principle (but not in practice) to recursively factor away the Riemann zeta zeros as they are computed?

Let: $$f_0(x)=\frac{\zeta (x)}{\sin \left(\frac{\pi x}{2}\right)}$$ and let the seed point be: $$s=\sqrt{-1}$$ which is the input into the limit: $$\rho_1=s+\lim\limits_{n \rightarrow \infty}\left(1-\...
Mats Granvik's user avatar
  • 1,133
0 votes
2 answers
215 views

Question about $\theta$ and the Riemann Hypthesis - reference request

It is well known that the Riemann Hypothesis implies the following: $|\theta(x) - x| = O(x^{1/2 + \epsilon})$ for all $\epsilon > 0$. where $\theta$ is the first Chebyshev function; that is, $\...
EGME's user avatar
  • 1,018
5 votes
0 answers
237 views

Estimating $\left| \sum_{n \leq x, n \neq p} \frac{\Lambda(n)}{n^{\sigma + it} \log n} \frac{\log x/n}{ \log n} \right|$ on RH

I am having some issue verifying Lemma 2 of K. Soundarajan's paper Moments of the Riemann Zeta function. It states the following: Assume RH. Let $T \leq t \leq 2T$, $2 \leq x \leq T^2$ and $\sigma \...
Lars's user avatar
  • 51
1 vote
1 answer
186 views

Asymptotics of cumulative Liouville function under RH versus simple random walk

The expectation values of the 1D simple random walk $S_n$ can be shown to have the asymptotic behavior of $$ \lim_{n\to\infty} \frac{a_n}{n^{1/2}} = \sqrt{\frac{2}{\pi}}, \tag{1}\label{1}$$ with $a_n =...
Raphael J.F. Berger's user avatar
5 votes
1 answer
2k views

Does the existence of a Landau-Siegel zero imply the existence of a complex zero off the critical line?

The question is in the title: can a Landau-Siegel zero be the only zero off the critical line for a Dirichlet L-function or does its existence imply the existence of a complex non trivial zero in the ...
Sylvain JULIEN's user avatar
0 votes
0 answers
189 views

On sum over non trivial zeroes of riemann zeta function

I wanted to know if there is an estimate or any closed form on the following partial sum series $$\sum_{n=1}^k \frac{1}{|\alpha_n||(\zeta'(\alpha_n))|}$$ Where "$\alpha_n$" runs over all non-...
47 47's user avatar
  • 11
3 votes
0 answers
128 views

What is the smallest sequence $a_k$ with nondecreasing $\frac{\sigma(a_k)-H_{a_k}}{\exp(H_{a_k})\log(H_{a_k})}$?

This is inspired by the Question on coefficient of $\exp(H_n).\log(H_n)$ in Lagarias equivalence of RH , an answer and some comments there. For $n\geqslant2$ denote $$ L(n):=\frac{\sigma(n)-H_n}{\exp(...
მამუკა ჯიბლაძე's user avatar
4 votes
2 answers
406 views

Question on coefficient of $\exp(H_n).\log(H_n)$ in Lagarias equivalence of RH

In page 197, Equivalents of the Riemann Hypothesis Vol 1, the following statement caught my eye There is an editorial comment in [102] that includes an observation by the GCHQ Problem Solving Group. ...
Prashanth Narasimha's user avatar
7 votes
1 answer
748 views

$\mathit{NP}$-hard statements which are $\mathit{NP}$-complete under the Riemann Hypothesis

$\newcommand\NP{\mathit{NP}}\newcommand\SAT{\mathit{SAT}}\newcommand\CH{\mathit{CH}}\newcommand\PSPACE{\mathit{PSPACE}}$Are there $\NP$-hard problems which are $\NP$-complete under the Riemann ...
Turbo's user avatar
  • 13.7k
8 votes
1 answer
832 views

A question on an equivalence of RH

In page 6, RH Equivalence 5.3. An equivalence of the Riemann Hypothesis says that $$\sum_{\rho} \frac{1}{|\rho|^2} =\sum_{\rho} \frac{1}{\rho (1{-}\rho)}= 2 + \gamma - \log 4\pi$$ where $\rho$ is ...
Beta's user avatar
  • 365
3 votes
1 answer
238 views

Robin's criterion, Goldbach's conjecture and upper bound for $r_{0}(n)$

This question is a follow-up to both About Goldbach's conjecture and Question in Proof of Hardy Ramanujan theorem about $\omega(n) =\sum_{p|n} 1$. Can one derive from Robin's criterion for RH an ...
Sylvain JULIEN's user avatar
1 vote
1 answer
595 views

Are the Riemann zeta zeros of the form $-\text{integer } i \pi +\log \left(\text{polynomial root}\right)$?

Let $\log(1),\log(2),\log(3),\log(4)...\log(n)$ be approximated by fractions generated by the truncated sums: $k=0$ $c=1$ $$\text{log1}=\sum_{n=0}^k \frac{0}{(1 n+1)^c}=0$$ $$\text{log2}=\sum _{n=0}^k ...
Mats Granvik's user avatar
  • 1,133
53 votes
6 answers
5k views

Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false

What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false? One interesting example, and the impetus for this question, is work in number ...
40 votes
1 answer
3k views

Connes–Consani's absolute geometry and Lurie's spectral algebraic geometry

Alain Connes and Caterina Consani seem to be currently working on "absolute algebraic geometry", which is a kind of "algebraic geometry over the sphere spectrum" (https://arxiv.org/...
Peter Bonart's user avatar
17 votes
0 answers
1k views

Colossally abundant numbers and the Riemann hypothesis

[This question followed up from a question on Math StackExchange.] Writing Robin's inequality for the Riemann hypothesis (RH) as $$\frac{\sigma(n)}{n \ln\ln n} < e^\gamma \;,$$ we can take ...
ntessore's user avatar
  • 219
28 votes
2 answers
3k views

What are some consequences of zero free strip of the Riemann zeta function?

A weaker version of the Riemann hypothesis is the claim that if $\zeta(s) = 0$ then $Re(s) \leq 1 - h$ for some constant $h> 0$. What would the consequences be of a result of this type?
Johnny T.'s user avatar
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