# 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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### 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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### 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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### 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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### 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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### '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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### 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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### 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?

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### Why is the Simple Zeros Conjecture said to be stronger than the Riemann Hypothesis?

Let the "Simple Zeros Conjecture (SZC)" be the statement that all zeros of the Riemann zeta function are simple.
I have often heard of the statement that the SZC is stronger than the Riemann ...

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### Exact formula for partial sums of Liouville function $L(n)$ (OEIS sequence A002819)

I am wondering if it is possible to get a useful exact formula, or at least some useful asymptotics, for the partial sums of the Liouville function (OEIS sequence A002819)
$$L(n)=\sum_{k=1}^n \lambda(...

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### Computability assertions for Riemann zeta zeros

While looking for information about the Riemann zeta function, I kept running into the claim that there is an algorithm to decide whether or not a zero of the function is off the half-line. Is this ...

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### Normal numbers, Liouville function, and the Riemann Hypothesis

This is a question about whether or not some number $\lambda^*$ is normal in base 2. More specifically, I am wondering if $\lambda^*$ is not normal. Proving it is normal would be next to impossible, ...

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### On primes of specified length and bit pattern

Denote $P(n,k)$ to be the number of primes between $2^n$ and $2^{n+1}-1$ having $k$ number of $1$s in its binary expansion between the $n+1$th binary digit and the least which is always $1$ if $n>1$...

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### Scaled Riemann zeta function with no zero in the critical strip

Update: I added $exp[i\theta_k(s)]$ in the definition of $\eta^*(s)$ to address some critical convergence issues. Thanks for the contributors who pointed to these issues.
Prime numbers are denoted as $...

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### Truncated Euler products, Dirichlet eta function, and convergence issues

Can you prove that the following series does not converge if $\frac{1}{2}<\sigma<1$, no matter how close to $1$ sigma is, and no matter how large $t>0$ is? The series is defined as
$$W(\sigma,...

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### The (current) obstructions for a cohomological interpretation of the Riemann zeta function

I am interested in the idea of a cohomological interpretation of the Riemann hypothesis (suggested by Deninger/Connes).
I am a beginner in étale cohomology, and I would like to ask the following
...

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### Prove that: $\sum _{c=1}^n \sum _{b=1}^n \sum _{a=1}^n \left(\left([b|c][b|a]\frac{\mu(b)b}{a}\right)-\frac{1}{a b\sqrt{c}}\right)<H_n+n$

In the OEIS there is the quote from Lowell Schoenfeld that the Riemann hypothesis is equivalent to:
$$|\psi(n) - n| < \sqrt{n} \log^2(n)$$
From the Euler Maclaurin formula one gets:
$$\sum _{c=1}^n ...

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### Can the Lagarias inequality be written as a "kernel inequality"?

The Lagarias inequality, which is equivalent to the Riemann hypothesis, is:
$$\sigma(n) \le H_n + \exp(H_n) \log(H_n) =:L(n)$$
for all natural numbers $n$, where $\sigma=$ sum of divisors, $H_n=n$-th ...

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### To which value does this infinite sum of power series coefficients converge?

Context:
In this and this paper, J. Arias de Reyna shows that the RH follows when:
$$1.2663935... \le \sum_{n=1}^\infty A_n^2 \le 1.2723669...$$
where $A_n$ is the coefficient in the following power ...

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### How differently would we model the distribution of primes if prime gap is larger?

Cramer's conjecture based on his random model provides prime gaps are bound by $O(\log^2p_n)$ where the gap is between $(n+1)$th and $n$th prime.
How differently would primes be modeled if gaps of $O(...

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### Axiom of determinacy as setting for studying rigs with $\operatorname{Aut}(\mathcal{M})\cong\operatorname {Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$?

I read years ago in the great French book "l'aventure des nombres" (Gilles Godefroy, Odile Jacob, 1997) that the axiom of determinacy was incompatible with the axiom of choice. The latter ...

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### Mertens Bound and the Riemann Hypothesis

Let $M(x)$ denote the Mertens function ($M(x)=\sum_{i=1}^{x}\mu(i)$ where $\mu(i)$ is the Möbius function) and let $\Lambda(i)$ denote the Mangoldt function ($\Lambda(i)$ equals $\log(p)$ if $i=p^{m}$ ...