Eigenvalue Density, Li’s Positivity, and the Critical Strip [PDF]
We rewrite the zero-counting formula within the critical strip of the Riemann zeta function as a cumulative density distribution; this subsequently allows us to formally derive an integral expression for the Li coefficients associated with the Riemann xi-
He, Y., Jejjala, V., Minic, D.
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Nearest neighbor spacing distributions for zeros of the real or imaginary part of the Riemann xi-function on vertical lines [PDF]
, 2014 Masatoshi Suzuki
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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 $p_n\left(x; \frac34,\frac34,\frac34,\frac34\right)$.
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On a positivity property of the Riemann ξ-function [PDF]
Acta Arithmetica, 1999 The functional equation for the Riemann zeta-function \(\zeta(s)\) is \(\xi(s) = \xi(1-s)\), where \[ \xi(s) := {\textstyle{1\over 2}}s(s-1)\pi^{-s/2}\Gamma( {\textstyle{1\over 2}}s)\zeta(s) = {\textstyle{1\over 2}}\prod_\rho {'} \left(1 - {s\over\rho}\right),\leqno(1) \] the product being taken over all complex zeros \(\rho\) of \(\zeta(s ...
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Dynamic behavior of the roots of the Taylor polynomials of the Riemann\n xi function with growing degree [PDF]
, 2016 Robert Jenkins, Ken D. T. -R. McLaughlin
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Proof of Riemann Conjecture Based on Contradiction between Xi-Function and Its Product Expression
, 2023 Chuanmiao Chen
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On properties of the Taylor series coefficients of the Riemann xi function at $s=\frac{1}{2}$ [PDF]
, 2019 Mario DeFranco
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Analysis of Voros criterion: what derivatives involving the logarithm of the Riemann xi-function at z=1/2 should be non-negative for the Riemann hypothesis holds true [PDF]
, 2014 Sekatskii Sergey
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Fourier Transforms of Positive Definite Kernels and the Riemann $$\xi $$ ξ -Function
Computational Methods and Function Theory, 2015 The purpose of this paper is to investigate the distribution of zeros of entire functions which can be represented as the Fourier transforms of certain admissible kernels. The principal results bring to light the intimate connection between the Bochner-Khinchin-Mathias theory of positive definite kernels and the generalized real Laguerre inequalities ...
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On a generalisation of the Riemann $\xi$-function
, 2021 It is known that we can construct the meromorphic function $Z_k(s)$ associated with the higher derivative of Hardy's $Z$-function. In this paper, we introduce the entire function derived from $Z_k(s)$, a generalisation of the Riemann $\xi$-function and prove some properties.
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