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Some identities related to Riemann zeta-function
Journal of Inequalities and Applications, 2016 It is well known that the Riemann zeta-function ζ ( s ) $\zeta(s)$ plays a very important role in the study of analytic number theory. In this paper, we use the elementary method and some new inequalities to study the computational problem of one kind of
Lin Xin
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Log-tangent integrals and the Riemann zeta function
Mathematical Modelling and Analysis, 2019 We show that integrals involving the log-tangent function, with respect to any square-integrable function on , can be evaluated by the harmonic series. Consequently, several formulas and algebraic properties of the Riemann zeta function at odd positive ...
Lahoucine Elaissaoui +1 more
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Series of Floor and Ceiling Functions—Part II: Infinite Series
Mathematics, 2022 In this part of a series of two papers, we extend the theorems discussed in Part I for infinite series. We then use these theorems to develop distinct novel results involving the Hurwitz zeta function, Riemann zeta function, polylogarithms and Fibonacci ...
Dhairya Shah +4 more
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A Probabilistic Proof for Representations of the Riemann Zeta Function
Mathematics, 2019 In this paper, we present a different proof of the well known recurrence formula for the Riemann zeta function at positive even integers, the integral representations of the Riemann zeta function at positive integers and at fractional points by means of ...
Jiamei Liu, Yuxia Huang, Chuancun Yin
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Extended Riemann Zeta Functions
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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Notes on the Riemann zeta-function-IV [PDF]
Hardy-Ramanujan Journal, 2000 In earlier papers of this series III and IV, poles of certain meromorphic functions involving Riemann's zeta-function at shifted arguments and Dirichlet polynomials were studied. The functions in question were quotients of products of such functions, and it was shown that they have ``many'' poles.
Balasubramanian, R. +3 more
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Fourier coefficients associated with the Riemann zeta-function
Karpatsʹkì Matematičnì Publìkacìï, 2016 We study the Riemann zeta-function $\zeta(s)$ by a Fourier series method. The summation of $\log|\zeta(s)|$ with the kernel $1/|s|^{6}$ on the critical line $\mathrm{Re}\; s = \frac{1}{2}$ is the main result of our investigation.
Yu.V. Basiuk, S.I. Tarasyuk
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Sonification of the Riemann Zeta Function [PDF]
Proceedings of the 25th International Conference on Auditory Display (ICAD 2019), 2019 The Riemann zeta function is one of the great wonders of mathematics, with a deep and still not fully solved connection to the prime numbers. It is defined via an infinite sum analogous to Fourier additive synthesis, and can be calculated in various ways.
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On the zeros of the Riemann zeta-function [PDF]
Proceedings of the American Mathematical Society, 1969 Let \(s=\sigma + it\) with \(\sigma\) and \(t\) both real, and put \[ R(s) = \pi^{-s/2} \Gamma(s/2) \zeta(s),\] where \(\zeta(s)\) denotes the Riemann zeta-function. Then, the functional equation of \(\zeta(s)\) can be written as \(R(s) =R(1-s)\). \textit{B. Berlowitz} [Acta Arith.
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Hamiltonian for the Zeros of the Riemann Zeta Function [PDF]
Physical Review Letters, 2017 5 pages, version to appear in Phys.
Brody, DC, Bender, CM, Müller, MP
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