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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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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3 votes
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518 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 ...
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3 votes
2 answers
368 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 ...
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4 votes
1 answer
233 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{\...
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1 vote
0 answers
161 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-\...
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0 votes
2 answers
196 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, $\...
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5 votes
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212 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 \...
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1 vote
1 answer
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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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4 votes
1 answer
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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 votes
0 answers
102 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-...
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3 votes
0 answers
115 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(...
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4 votes
2 answers
354 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. ...
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7 votes
1 answer
690 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 ...
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8 votes
1 answer
596 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 ...
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3 votes
1 answer
202 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 ...
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1 vote
1 answer
533 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 ...
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  • 1,017
50 votes
6 answers
4k 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 ...
28 votes
1 answer
2k views

Connes'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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17 votes
0 answers
978 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 ...
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27 votes
2 answers
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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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8 votes
2 answers
2k views

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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4 votes
2 answers
515 views

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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7 votes
0 answers
206 views

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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11 votes
1 answer
1k views

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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1 vote
0 answers
112 views

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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-4 votes
1 answer
352 views

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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2 votes
1 answer
271 views

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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9 votes
1 answer
734 views

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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1 vote
0 answers
136 views

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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4 votes
1 answer
264 views

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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4 votes
0 answers
180 views

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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0 votes
1 answer
219 views

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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0 votes
0 answers
257 views

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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2 votes
0 answers
293 views

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 $i=p^{m}$ ...
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3 votes
0 answers
191 views

Lower bound of the modulus $|\eta(s)|$ of the Dirichlet Eta function if $0.6 < \Re(s) < 0.9$

Let $s=\sigma + it$, with $0.6 < \sigma < 1$ and $\sigma=\Re(s)$. I am trying to get good enough approximations for $\eta(s)$, hoping something useful might come out of it. I stumbled upon a ...
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3 votes
0 answers
332 views

On RH in the Clay Institute list

As everybody knows, the Riemann Hypothesis is one of the problems of the millenium raised by the Clay Institute. Looking at the "official formulation" of various problems, say for instance ...
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3 votes
1 answer
131 views

A Hadamard product representation for Keiper's $\tau$-function?

In this paper J.B. Keiper defined the following function: $$\tau_k = \sum_{j=1}^k (-1)^j\,{k-1 \choose j-1} \sigma_{j+1} \qquad k \ge 1, k \in \mathbb{N} \tag{1}$$ where $\displaystyle\sigma_r = \sum_{...
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0 votes
1 answer
251 views

On some property of the zeros of $\zeta(s)$ in the complex plane

This property is rather elementary, and not at all specific to $\zeta$, so I am wondering if it has any value in studying the zeros of the Riemann zeta function in the critical strip. It is a well ...
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26 votes
1 answer
2k views

Riemann's attempts to prove RH

I read somewhere that Riemann believed he could find a representation of the zeta function that would allow him to show that all the non-trivial zeros of the zeta function lie on the critical line. I ...
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1 vote
0 answers
526 views

Why didn't Robin prove the Riemann Hypothesis?

I'm reading Robin's paper, ''Grandes valeurs de la fonction somme des diviseurs et hypoth`ese de Riemann,'' J. Math. Pures Appl. (9) 63 (1984). In particular, Lemma 5 states that $\prod_{p\leq P} (1-p^...
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21 votes
1 answer
2k views

More mysteries about the zeros of the Riemann zeta function

Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$. Update on 1/5/2020: I added the section "more interesting ...
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3 votes
1 answer
366 views

Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors?

Robin's inequality $$\sigma_1(n)<e^\gamma n\log\log n$$ at integers $n>5040$ provides necessary and sufficient condition for Riemann Hypothesis where $\sigma_1(n)=\sum_{d|n}d$ is sum of divisors ...
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2 votes
2 answers
756 views

Prove that the real part of this limit converges to $\frac{1}{2}$

Let: $s= 1/3 + 14i$ Prove that the real part of this limit converges to $\frac{1}{2}$: $$\Re\left(\lim_{n \rightarrow \infty}\left(\frac{1}{1-\frac{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\...
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  • 1,017
2 votes
0 answers
567 views

An interesting sequence of numbers arising from the Riemann hypothesis

A very good coincidence occurred today with me. While just plotting random functions in Mathematica, I entered this command: ...
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0 votes
1 answer
311 views

Where can I find the problem by Lagarias?

Jeffrey Lagarias proved, unconditionally, that: $$ \sigma(n)<H_n+2\exp(H_n)\log(H_n)\qquad n>1 $$ This was posed as a problem in: J. C. Lagarias, Problem 10949: A generous bound for divisor ...
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3 votes
1 answer
167 views

Error term for the summatory function of $k$-free numbers indicator and RH

I started to read this preprint: https://arxiv.org/abs/2010.03696 In it, the author states that $\sum_{n\leq x}\mu_{k}(n)=\zeta(k)^{-1}x+O(x^{1/k})$ and that under RH, the exponent in the error term ...
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2 votes
1 answer
570 views

$\frac{\sigma(n)}{n} < e \ln \ln (n)$ is true?

In Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187–213 (pdf) we find the following result: If the Riemann hypothesis is true ...
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7 votes
1 answer
449 views

Riemann hypothesis for exponential sum

Recently I've heard about the Riemann hypothesis for one-variable exponential sums, which states as For a polynomial $f\in\mathbb{F}_{p^k}[x]$ of degree $d$ and a character $\chi$ of $(\mathbb{F}_{p^...
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3 votes
3 answers
437 views

Show that the ratio of limits converges to the nearest Riemann zeta zero except when the ratio is a singularity

Let $h(s,n)$ be: $$h(s,n)=\lim_{c\to 1} \, \frac{(-1)^{n-2}}{(n-2)!}\zeta (c)^{n-2} \sum _{k=1}^{n-1} \frac{(-1)^{k-1} \binom{n-2}{k-1}}{\zeta ((c-1) (k-1)+s)}$$ and let $g(s,n)$ be: $$g(s,n)=\lim_{c\...
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  • 1,017
1 vote
0 answers
152 views

Is $T(n)=\sum_{k=1}^{n}\frac{\lambda(k)\Lambda(k)}{k} \geq 0$ and what is the upper bound of $T(x)=\sum_{n\leq x} \lambda(n)\Lambda(n) $?

Let $\Lambda(n)$ denote the von Mangoldt function: $\Lambda(n)=\log p$ when $n=p^e$ is a prime power ($e\ge 1$) and $\Lambda(n)=0$ otherwise. and $\lambda(n)$ be Liouville Function, , I'm interested ...
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