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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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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PT Symmetry and the Riemann Hypothesis
Recently there have been articles in Quanta, in Science Alert, and at phys.org among others, on possible recent progress toward the Hilbert-Polya conjecture, which implies the Riemann Hypothesis. The ...
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On a revised quantum Riemann hypothesis
This post provides a revision of the disproved quantum Riemann hypothesis proposed 2 years ago in this post, where you can refer to have more details about the motivations, the notations and the ...
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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 ...
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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 $...
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Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$
If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{...
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Example of sequence of graphs which satisfy the Riemann hypothesis or the prime number theorem?
Let us look at the sequence of bipartite graphs $G_n = (V_n, E_n)$ where $V_n = A_n \cup B_n$ defined in this quesiton: Why is this bipartite graph a partial cube, if it is? .
The shortest path ...
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Toward a cyclotomic Riemann hypothesis
For an integer $n \ge 3$, consider the function $$u(n) = \frac{\sigma(n)}{n \log \log n}$$ with $\sigma$ the divisor function. Now consider the sequence (bounded below and decreasing) $$v_n = \sup_{m&...
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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 ...
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Riemann hypothesis in Zilber's field
Question. What is known about the situation (truth or falsity) of Riemann hypothesis in the Zilber's field?
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Questions on de Branges' work on the Riemann hypothesis
According to Wikipedia, Louis de Branges de Bourcia has obtained some notable
results, such as a proof of the Bieberbach conjecture in 1985, which is now
known as de Branges' theorem. Initially, his ...
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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
$$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
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On Riesz criteria for Riemann hypothesis:
Marcel Riesz defined a function :
$R(x) = \sum_{n=1}^\infty \frac {(-1)^n x^n} {\zeta(2n)\Gamma(n)}$
The Riemann hypothesis holds if $R(x)= O( x^{1/4 + {\varepsilon}}$)
For any $\varepsilon$
We have ...
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Numerical Evidence for Grand Riemann Hypothesis?
Let $L(s)$ be an $L$-function coming from Hecke characters or automorphic forms (e.g. modular form on GL(2), Maass form on GL(2), and higher-rank analogues).
Is there any numerical evidence for ...
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Do all complex zeros in the strip of $\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$ lie on the critical line?
Numerical evidence suggests that the complex zeros of:
$$f(s):=\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$$
all reside on the line $\Re(s)=\frac12$, except for a finite few outside ...
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Order of magnitude of extremely abundant numbers and RH
I have always been intrigued by the fact that Riemann's hypothesis is equivalent to the assertion (you can find the scanned paper here) that the inequality $$\frac{\sigma(n)}n<e^\gamma \log\log n \...
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Riemann hypothesis for the Hecke operators and modular forms
Let $f(z) = \sum_{n=1}^\infty a(n) e^{2i \pi nz}$ be an eigenform of $S_k(\Gamma_0(N))$. Since the Hecke operator acts by $T_p f = a_p f$ the Riemann hypothesis for $f$'s L-function is
$$ \!\!\! \!\...
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Computability assertions for Riemann zeta zeros
While looking for information about the Riemann zeta function, I kept running into the claim that there is an algorithm to decide whether or not a zero of the function is off the half-line. Is this ...
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What is known about "almost orthogonal vectors"?
Motivation:
Suppose we have a kernel $k(a,b)$ defined over the natural numbers.
Then by the Moore–Aronszajn theorem, we can embedd the natural number $a$ in some Hilbert space $\mathbb{H}$, which we ...
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How to assess the influence of a specific term in this telescoping series for $\zeta(s)$?
I like to expand on this (unanswered) MSE question.
Take the following, nicely symmetrical, telescoping series for $\zeta(s)$:
$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(1+\sum _{n=1}^{\...
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Is there a conjectured uniform Lindelof hypothesis for Hurwitz zeta functions
Consider $\zeta(s, a) = \sum_{n=1}^{\infty} (n+a)^{-s}$ (alternatively, consider its functional equation Dirichlet series $\sum_{n=1}e(a n) n^{-s}$). What is the expected growth-rate of $\zeta(1/2 + ...
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Are all complex zeros of $\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$ on the critical line for all $z \lt 1$?
Numerical evidence suggests that all complex zeros residing in the critical strip $0 < \Re(s) < 1$ of:
$$\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$$
are on the ...
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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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Are there infinitely many zeros of $\chi(s)+ \dfrac{2^{s}- 2^{2s-1}}{2^{s}-1}$ on the critical line?
Take $\chi(s)= 2^s\,\pi^{s-1}\,\sin\left(\frac{\pi\,s}{2}\right)\,\Gamma(1-s)$, so that $\zeta(s)=\chi(s)\,\zeta(1-s)$.
The zeros of $\chi(s)=-1$ and the non-trivial zeros $\rho$ of $\zeta(s)$, seem ...
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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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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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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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Is there a hidden symmetry in the prime numbers distribution?
Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime.
Let'...
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Possible weakening of Robins criterion for RH
I hope 2015 doesn't start with gross nonsense question for me.
Robins equivalence for RH is
$$ \frac{\sigma(n)}{n \log\log n} < e^{\gamma}\qquad(1)$$
for $n \ge 5041$. We have
$ \limsup \frac{\...
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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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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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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
$$
L(n):=\frac{\sigma(n)-H_n}{\exp(...
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Lower bound of the modulus $|\eta(s)|$ of the Dirichlet Eta function if $0.6 < \Re(s) < 0.9$
Let $s=\sigma + it$, with $0.6 < \sigma < 1$ and $\sigma=\Re(s)$. I am trying to get good enough approximations for $\eta(s)$, hoping something useful might come out of it. I stumbled upon a ...
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Largest observed value of $S(t)$
Let $S(t)$ be the deviation of the number of zeros of the Riemann zeta function up to height $t$ from the expectation.
What is the largest observed value of $S(t)$ today?
Here is a quote from a ...
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A question about Lagarias inequality
Let $|g|=\min(g,n-g)=$Lee-norm on $\mathbb{Z}/(n)=$ word length on cyclic group $C_n$ with respect to generating set $S=\{\pm 1\}$. Let $H^{L}(C_n) := \sum_{g \in C_n} \frac{1}{|g|+1}$, which as one ...
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Does the proof of Conjectures B and D of Hardy and Littlewood have any implication on the generalized Riemann hypothesis they used?
Does the proof of Conjectures B and D of Hardy and Littlewood have any implication on the generalized Riemannn hypothesis they used?
In their paper,
Some problems of 'Partitio numerorum'; III - On ...
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Are the complex zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$ all on the critical line for $a \lt 0, a \ge 1$?
With $s \in \mathbb{C}, a \in \mathbb{R}$,
numerical evidence strongly suggests that the complex zeros in the critical strip of:
$$\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$$...
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Definite integral of $\zeta(s)$ over the critical strip
Take the following definite integral:
$$f(s):=\int_s^{1-s} \zeta(x) \mathrm{d} x$$
with $s \in \mathbb{C}$, $s=\sigma \pm ti$, $0<\sigma<1$ and $t,\sigma \in \mathbb{R}$.
The graph of $|f(s)|...
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Characterizing essential singularities
In the paper Picture of an essential singularity, an analogy is made between the multipolar moments of infinitesimal charge distributions and the lines of constant modulus/argument around an essential ...
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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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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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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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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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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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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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Explicit formula for $n$th prime in terms of Riemann zeros:
We all know there exists an 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?
Or any other ...
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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^{...
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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 ...
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On a sequence of L-functions having same zeros in critical strip and GRH
I had an idea on GRH involving a sequence of L-functions having same zeros, then at one step I need a bound on these function and I wonder if this bound is in fact not as hard as GRH itself ?
Let's ...
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Are all zeros of $\xi(a\,s) \pm \xi\left(a\,(1-s)\right)$ on the critical line for $\forall a \in \mathbb{R}/0$?
This question expands on this one and seems to have a stronger result.
Take the Riemann $\xi$-function $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. We ...