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Theta and Riemann xi function representations from harmonic oscillator eigensolutions [PDF]
Physics Letters A, 2006 From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical zeta function. A
Mark W. Coffey
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Jensen polynomials for the Riemann xi-function [PDF]
Advances in Mathematics, 2022 We investigate Riemann's xi function $ξ(s):=\frac{1}{2}s(s-1)π^{-\frac{s}{2}}Γ(\frac{s}{2})ζ(s)$ (here $ζ(s)$ is the Riemann zeta function). The Riemann Hypothesis (RH) asserts that if $ξ(s)=0$, then $\mathrm{Re}(s)=\frac{1}{2}$. Pólya proved that RH is equivalent to the hyperbolicity of the Jensen polynomials $J^{d,n}(X)$ constructed from certain ...
Michael Griffin +5 more
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On solution of fractional kinetic equation involving Riemann xi function via Sumudu transform
International Journal of Mathematics for Industry, 2023 Several significant questions of mathematics and mathematical physics have been effectively explained and answered through the use of fractional kinetic equations containing special functions.
Mulualem Aychluh
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Higher order Turán inequalities for the Riemann $\xi$-function [PDF]
Proceedings of the American Mathematical Society, 2010 As usual let \(\zeta(z)\) denote the Riemann \(\zeta\)-function, \(\Gamma(z)\) denote the gamma function, and recall the Riemann \(\xi\)-function defined by \[ \xi(iz)=\frac12(z^2-1/4)\pi^{-z/2-1/4}\Gamma(z/2+1/4)\zeta(z+1/2). \] It is known that the Turán inequalities \(\gamma_k^2-\gamma_{k-1}\gamma_{k+1}\geq 0~(k\in\mathbb{N})\) are valid for the ...
Dimitar K. Dimitrov, Fábio Lucas
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On a Bessel function series related to the Riemann xi function [PDF]
, 2023 Revised
Alexander E. Patkowski
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A family of deformations of the Riemann xi-function [PDF]
Acta Arithmetica, 2013 We introduce a family of deformations of the Riemann xi-function endowed with two continuous parameters. We show that it has rich analytic structure and that its conjectural (mild) zero-free region for some fixed parameter is a sufficient condition for the Riemann hypothesis to hold for the Riemann zeta function.
Masatoshi Suzuki
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Real-rooted Pólya-like approximations to the Riemann Xi-function [PDF]
, 2014 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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A Note on the $S_2(\delta)$ Distribution and the Riemann Xi Function [PDF]
Electronic Communications in Probability, 2014 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Dmitry Ostrovsky
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The Integral of the Riemann xi-function [PDF]
, 2011 This paper studies the integral of the Riemann xi-function. More generally, it studies a one-parameter family of functions given by Fourier integrals and satisfying a functional equation. Members of this family are shown to have only finitely many zeros on the critical line, with the integral of the Riemann xi-function having exactly one zero on the ...
Jeffrey C. Lagarias, David Montague
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