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Extended Riemann Zeta Functions [PDF]
Rocky Mountain Journal of Mathematics, 2001 In this paper two extensions of the Riemann zeta-function are presented. Denoting the real part of the complex variable \(s\) by \(\sigma\), these extensions are \[ \zeta_{b}(s) = {1\over \Gamma(s)}\int_{0}^{\infty}t^{s-1}(e^{t}-1)^{-1}e^{-b/t} dt \quad \quad (b>0; b=0, \sigma >1), \] and \[ \zeta_{b}^{*}(s) = {1\over \Gamma(s)(1-2^{1-s})}\int_{0 ...
Chaudhry, M. Aslam +4 more
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Riemann’s zeta function and beyond [PDF]
Bulletin of the American Mathematical Society, 2003 In recent yearsLL-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving the analytic continuation and functional equations ofLL-functions: the method of integral representations, and the method of Fourier expansions of ...
Gelbart, Stephen S., Miller, Stephen D.
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International Journal of Mathematics and Mathematical Sciences, 2004 We consider the modified q‐analogue of Riemann zeta function which is defined by , 0 < q < 1, s ∈ ℂ. In this paper, we give q‐Bernoulli numbers which can be viewed as interpolation of the above q‐analogue of Riemann zeta function at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers ...
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A Bicomplex Riemann Zeta Function
Tokyo Journal of Mathematics, 2004 The author uses a commutative generalization of complex numbers, called bicomplex numbers, to introduce a holomorphic Riemann zeta function of two complex variables, which satisfies the complexified Cauchy-Riemann equations. Moreover, the author establishes a bicomplex Riemann hypothesis which is equivalent to the complex Riemann hypothesis of one ...
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Inequalities for Riemann’s zeta function
Portugaliae Mathematica, 2009 Let ζ and Λ be the Riemann zeta function and the von Mangoldt function, respectively. Further, let c > 0
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, 2009 Global mapping properties of the Riemann Zeta function are used to investigate its non trivial zeros.
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, 2019 El propósito de este trabajo de grado es dar una breve introducción a la función Zeta de Riemann, explorando sus principales propiedades como lo son su continuación analítica, fórmula de reflexión y estudiar los ceros de dicha función. ; The purpose of this thesis is to give a brief introduction to the Riemann Zeta function, explaining the most ...
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Extreme values of derivatives of the Riemann zeta function. [PDF]
Mathematika, 2022 Yang D.
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Mathematische Zeitschrift, 1986 Riemann's ''second proof'' of the functional equation for \(\zeta\) (s) depends on the inversion formula for a classical theta-function, and consists in proving for \(\pi^{-s/2} \Gamma (s/2) \zeta (s)\) an expression which remains invariant if s is replaced by \(1-s.\) Generalizing this approach, the authors introduce certain polynomials \(p_ j(s ...
Bump, Daniel, Ng, Eugene K.-S.
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Characteristic polynomials, spectral-based Riemann-Zeta functions and entropy indices of n-dimensional hypercubes. [PDF]
J Math Chem, 2023 Balasubramanian K.
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