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.
267 questions
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Can Weil positivity be realized as an adelic norm-square identity? [closed]
I am interested in the Weil positivity criterion arising from the explicit formula for the Riemann zeta function. More precisely, I have in mind the quadratic form usually written as
$$
Q(g) \geq 0,
$$...
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Which AI models or architectures can learn and extend the covering algorithm of this partition-based framework to address the Riemann Hypothesis? [closed]
(I am not announcing new results or asking to verify a proof or a preprint. This post is not a a request to check the work for correctness or an announcement of results. I hope this question meets all ...
17
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1
answer
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Why does a truncated Euler product do so well at detecting peaks in the Riemann zeta function on the critical line?
Here's a phenomenon I'm curious about. The Euler product
$$ \prod_{p \textrm{ prime}} (1 - p^{-s})^{-1} $$
converges to the Riemann zeta function $\zeta(s)$ for $\text{Re}(s) > 1$. It
does not ...
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On Chowla's conjecture on primes in AP
Pomerance in A Note on the Least Prime in an Arithmetic Progression states under $GRH$ Chowla shows we have $$P(K)\ll k^{2+\epsilon}$$ where $$P(k)=\max_{l\in\{1,\dots,p-1\}:(k,l)=1}p(k,l)$$ where $p(...
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Is there a natural topos where the Riemann hypothesis is provable or disprovable?
While constructive logic is compatible with classical logic and is sufficient to develop almost all important theorems from classical complex analysis, constructive is also compatible with axioms that ...
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1
answer
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The Riemann hypothesis and the parity of prime divisors with multiplicity
Motivation. In his interesting and concise answer a recent question how to "best" explain the Riemann hypothesis to a general audience, user GH from MO writes:
Roughly speaking, the ...
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0
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1
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Is phase $S(T)$ of Riemann Zeta function jumping maximum by one for small increase of $T$?
The number of non-trivial zeros of the $\zeta$ function is strongly coupled to
the hypothetical number of zeros outside of the critical line that are
counter-examples for the Riemann Hypothesis. Hence,...
7
votes
2
answers
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Where is paper proving asymptotic growth of Nicolas criterion for Riemann Hypothesis?
Nicolas has shown Nicolas result that if
\begin{equation}\label{Gk}
G(k)=G_0(k)-{\rm e}^{\gamma}\ln\ln N_k>0,
\end{equation}
for all $k\ge 2$, the Riemann Hypothesis is true.
\begin{equation}
...
41
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3
answers
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Best formulation of Riemann hypothesis for a general audience
There are many equivalent ways to state the Riemann Hypothesis. I'm looking for a statement that is mathematically precise and yet at the same time as accessible as possible to a general audience. The ...
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Why is it hard to get a zero-free half-plane $\Re s>\tfrac12$ for $\zeta(s)$ via Rouché using the Maclaurin (Euler–Maclaurin) formula?
I start from the classical “Maclaurin” form of Euler–Maclaurin for every integer $n\ge2$:
$$
\zeta(s)
=\frac{1}{s-1}+\frac12+\sum_{k=2}^{n} B_k\,\frac{s(s+1)\cdots(s+k-2)}{k!}
-\frac{s(s+1)\cdots(s+n-...
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If the Riemann hypothesis is false, how short is the proof?
If the Riemann hypothesis is false, it is provably false: there would be a nontrivial zero off the critical line, and a finite computation can prove the existence of the zero.
How tightly can we bound ...
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Applications of Bang's lemma as in the solution to Tarski's problem to a number theoretic problem?
Inspired by this question and answer, I want to apply Bang's lemma to a number theoretic setting:
Bang's lemma for p.d. kernels: Let $k : X \times X \rightarrow \mathbb{R}$ be a positive definite ...
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1
answer
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Is Re(m)=Re(n) the only condition for $\displaystyle \int_0^{\infty} \frac{t^m-t^n}{e^t-1} \, dt = 0$ to hold? [closed]
For such an integral: $$\displaystyle \int_0^{\infty} \frac{t^m-t^n}{e^t-1} \, dt = 0$$
Given that m and n are constants and we suppose that m and n are equal, it can clearly be observed that the ...
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votes
1
answer
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Power series expansion of the inverse of the function $L(x) = x+\exp(x) \log(x)$?
I am searching for the power series expansion of the inverse of the function
$L(x) = x+\exp(x) \log(x)$
This function occurs at the Robin-Lagarias inequality equivalent to RH:
$$\sigma(n) \le L(H_n)$$
...
2
votes
0
answers
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Dirichlet series from second differences of recursively summed prime gaps
I'm investigating a Dirichlet series built from a recursively summed and differenced sequence of prime gaps.
$\text{Let } g_n = p_{n+1}-p_n$, denote the prime gaps. From these, construct:
$$S_0(n)=g_n,...
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