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Results tagged with riemann-zeta-function
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user 13625
The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.
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Is the upper bound on $H_{1}$ a decreasing function of the proportion of critical zeros of Z...
This question stems from https://arxiv.org/abs/2411.19762 and the numerical observation that the best unconditional upper bound for $H_{1}:=\lim\inf_{n\to\infty}p_{n+1}-p_{n}$, namely $H_{1}^{\flat}=2 …
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Functional equation and/or growth estimates for a shifted L function
This preprint by Kaczorowski and Perelli may contain the pieces of information you're looking for: https://arxiv.org/abs/1911.10497
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Do we believe that the distribution of spacings of successive critical zeros of zeta is log-...
Let $\gamma^{+}(T)$ be the imaginary part of the critical zero of $\zeta$ closest to $1/2+iT$ with $\gamma^{+}(T)\ge T$ and define similarly $\gamma^{-}(T)$ with a reversed inequality. Let $g(T)$ be t …
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What is the best known upper bound for $( \gamma_{n+1}-\gamma_{n})\max_{\{T\in(\gamma_{n},\g...
For $ n $ a positive integer, denote by $ L(n) : =\gamma_{n+1}-\gamma_{n} $ with $ \gamma_{n} $ the imaginary part of the $ n $-th critical zero of the Riemann zeta function and by $ M(n) : =\max …
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On the real part of the Riemann zeta function inside the critical strip
See https://www.researchgate.net/publication/321187136_Pair_Correlation_of_Zeros_of_the_Real_and_Imaginary_Parts_of_the_Riemann_Zeta-Function
where the authors investigate the behavior of the real an …
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Has there been further work on Bender-Brody-Müller approach to RH?
Earlier this year (April 4, 2017), a seemingly tantalizing approach of the Riemann Hypothesis based on ideas dating back to Hilbert and Pólya by Bender, Brody and Müller was made publicly available. I …
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Is the number of real values of Zeta on the critical line up to some ordinate known?
The famous plot of $\zeta(1/2+it)$ for real $t$ seems to show that this function gets a non zero real value exactly once between two consecutive Riemann zeros. Moreover, letting $\rho_{i}$ the $i$-th …
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Asymptotic number of zeros for Dirichlet series with functional equation
See the theorem 2.15 in the survey of the Selberg class by Li ZHENG.
https://www.researchgate.net/publication/265103975_A_CONCISE_SURVEY_OF_THE_SELBERG_CLASS_OF_L-FUNCTIONS
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Do we know an upper bound for the number of possible real parts of the non trivial zeroes of...
Let $n_{\zeta}$ denote the number of possible real parts for the non trivial zeroes of the Riemann Zeta function. RH is equivalent to $n_{\zeta}=1$, and the symmetry arising from the functional equati …
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Can Voronin's universality theorem be used to show that $\sigma\circ\zeta=\zeta\circ\sigma$ ...
I apologize for answering my own question, but it has turned out that the statement I consider can actually be proved without using Voronin's theorem.
Here comes an excerpt from an article of mine en …
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what would be the consequences on the distribution of primes of $\Lambda=\infty$?
It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where $t_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the Rieman …
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Do we know a lower bound for the number of critical zeros of the Riemann zeta-function with ...
If I'm not mistaken, the imaginary parts of the critical zeros of the Riemann Zeta function are conjectured to be linearly independent over $\mathbb{Q}$, but I think we're very far from proving such a …
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Can Voronin's universality theorem be used to show that $\sigma\circ\zeta=\zeta\circ\sigma$ ...
Let $\sigma$ be a field automorphism of $\mathbb{C}$ that commutes with the Riemann Zeta function. Can we use Voronin's universality theorem to prove that $\sigma$ is necessarily continuous?
Thanks in …
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What is known about the set $S$ of couples of rationals $(q,q')$ such that $\zeta(q+iq')$ is...
The question is the title. For example, if we could show that $S$ is finite, then this would entail that every large enough integer $n$ is such that $\zeta(2n+1)$ is irrational and that, under RH, alm …
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Do we know an upper bound for the multiplicity of the non-trivial zeros of Zeta?
In Are the nontrivial zeros of the Riemann zeta simple?, I asked whether it was known that all non-trivial zeros of the Riemann Zeta function were simple or not. It appears that such a proof is missin …
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