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Orthogonal polynomial expansions for the Riemann xi function
, 2019 We study infinite series expansions for the Riemann xi function $ (t)$ in three specific families of orthogonal polynomials: (1) the Hermite polynomials; (2) the symmetric Meixner-Pollaczek polynomials $P_n^{(3/4)}(x; /2)$; and (3) the continuous Hahn polynomials $p_n\left(x; \frac34,\frac34,\frac34,\frac34\right)$.
Dan Romik
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Phase variation of the Euler product, and, angular momentum of the Riemann, and, Dirichlet Xi functions [PDF]
The concept of angular momentum is used to find new RH equivalence statements, and, generalize some known results from Riemann to Dirichlet primitive Xi ...
Giovanni Lodone
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On the zeros of Riemann's Xi Function
, 2023 We consider Riemann's Xi function $ξ(s)$ which is evaluated at $s = \frac{1}{2} + σ+ i ω$, given by $ξ(\frac{1}{2} + σ+ i ω)= E_{pω}(ω)$, where $σ, ω$ are real and compute its inverse Fourier transform given by $E_p(t)$. We study the properties of $E_p(t)$ and a promising new method is presented which could be used to show that the Fourier Transform of
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Real-rooted Pólya-like approximations to the Riemann Xi-function
, 2015 The Riemann $ (z)$ function admits a Fourier transform of a even kernel $ (t)$. The latter is related to the derivatives of Jacobi theta function $ (z)$, a modular form of weight $1/2$. P lya noticed that when $t$ goes to infinity, $e^t$ goes to $e^t+ e^{-t}=2\cosh t$.
Yaoming Shi
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Riemann Xi function and equivalent of Riemann Hypothesis
, 2021 Equivalent of ...
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Riemann Hypothesis proof using Hadamard product and Riemann xi function
, 2021 Easy proof.
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Power series expansions of Riemann’s $\xi$ function [PDF]
Mathematics of Computation, 1992 High-precision values of the coefficients of power series expansions about the point \(s=1\) are obtained for the Riemann \(\xi\) function, its logarithmic derivative \(\xi'(s)/\xi(s)\), and the related functions \(\xi'(1/s)/\xi(1/s)\) and \(\log(2\xi(1/s))\).
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Restricted Tweedie stochastic block models
Canadian Journal of Statistics, EarlyView.Abstract The stochastic block model (SBM) is a widely used framework for community detection in networks, where the network structure is typically represented by an adjacency matrix. However, conventional SBMs are not directly applicable to an adjacency matrix that consists of nonnegative zero‐inflated continuous edge weights.
Jie Jian, Mu Zhu, Peijun Sang
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Riemann Hypothesis proof using Logarithmic derivative of Riemann Xi function
, 2021 easy ...
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