Results 11 to 20 of about 66,497 (169)
A monotonicity property of Riemann’s xi function and a reformulation of the Riemann hypothesis [PDF]
Periodica Mathematica Hungarica, 2010 4 pages, published ...
Jonathan Sondow, Cristian Dumitrescu
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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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Zeros of Jensen polynomials and asymptotics for the Riemann xi function [PDF]
Research in the Mathematical Sciences, 2021 The classical criterion of Jensen for the Riemann hypothesis is that all of the associated Jensen polynomials have only real zeros. We find a new version of this criterion, using linear combinations of Hermite polynomials, and show that this condition holds in many cases.
Cormac O’Sullivan
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The Integral of the Riemann xi-function
, 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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On a Bessel function series related to the Riemann xi function
, 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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On the harmonic continuation of the Riemann xi function
, 2023 We generalize the harmonic continuation of the Riemann xi-function to the $n$-dimension case, to obtain the solution to the Dirichlet problem on $\mathbb{R}_{+}^{n+1}.$ We also provide a new expansion for the harmonic continuation of the Riemann xi-function using an expansion given by R.J. Duffin.
Alexander E. Patkowski
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A Note on the $S_2(\delta)$ Distribution and the Riemann Xi Function [PDF]
Electronic Communications in Probability, 2014 The theory of $S_2(\delta)$ family of probability distributions is used to give a derivation of the functional equation of the Riemann xi function. The $\delta$ deformation of the xi function is formulated in terms of the $S_2(\delta)$ distribution and shown to satisfy Riemann's functional equation.
Dmitry Ostrovsky
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On special Riemann xi function formulae of Hardy involving the digamma function [PDF]
Acta Scientiarum Mathematicarum, 2021 We consider some properties of integrals considered by Hardy and Koshliakov, and which have also been further extended recently by Dixit. We establish a new general integral formula from some observations about the digamma function. We also obtain lower and upper bounds for Hardy's integral through properties of the digamma function.
Alexander E. Patkowski
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ON A TEMPERED XI FUNCTION ASSOCIATED WITH THE RIEMANN XI FUNCTION
Fractals, 2023 In this paper, we propose a tempered xi function obtained by the recombination of the decomposable functions for the Riemann xi function for the first time. We first obtain its functional equation and series representation. We then suggest three equivalent open problems for the zeros for it. We finally consider its behaviors on the critical line.
Xiao‐Jun Yang
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