# 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), ...

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2
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### 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(...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $ℜs> 1/ 2$.&...

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### 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^{\...

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### 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 ...

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### 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\, \...

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### 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)}{\...

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0
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### 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 ...

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### 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 ...

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### 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 ...

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### 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}{...

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0
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?
...

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### 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<.....

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1
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### 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 $F$ ...

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1
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### '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 $R= \{\sigma+i t\mid |\sigma -\frac{1}{2}| \leq \Phi(t) \}$...

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### 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/...

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### 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 ...

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### 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 ...

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### 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\...

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### 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(\...

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### 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 $C$ ...

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### 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 ...

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2
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### 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 ...

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### 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{\...

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### 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-\...

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### 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, $\...

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### 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 \...

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### 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 =...

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### 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 ...

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0
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### 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-...

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### 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(...

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### 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. ...

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### $\mathit{NP}$-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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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/...

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### 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 ...

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### 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?