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An explicit formula for Pi(x) in the form of a sum over the Non-trivial zeros of the Riemann Zeta function  [PDF]
Jose Javier Garcia Moreta
Mathematics , 2006,
Abstract: In this paper we discuss a method to express the Prime counting function as a "sum" over Non-trivial zeros of Riemann Zeta function, using techniques from Analytic Number Theory, also we apply our results to the sum over primes of any function Sum(p
Determinants of Laplacians, the Ray-Singer Torsion on Lens Spaces and the Riemann zeta function  [PDF]
Charles Nash,Denjoe O' Connor
Physics , 1992, DOI: 10.1063/1.531134
Abstract: We obtain explicit expressions for the determinants of the Laplacians on zero and one forms for an infinite class of three dimensional lens spaces $L(p,q)$. These expressions can be combined to obtain the Ray-Singer torsion of these lens spaces. As a consequence we obtain an infinite class of formulae for the Riemann zeta function $\zeta(3)$. The value of these determinants (and the torsion) grows as the size of the fundamental group of the lens space increases and this is also computed. The triviality of the torsion for just the three lens spaces $L(6,1)$, $L(10,3)$ and $L(12,5)$ is also noted. (postscript figures available as a compressed tar file)
Ray-Singer Torsion, Topological field theories and the Riemann zeta function at s=3  [PDF]
Charles Nash,Denjoe O' Connor
Physics , 1992,
Abstract: Starting with topological field theories we investigate the Ray-Singer analytic torsion in three dimensions. For the lens Spaces L(p;q) an explicit analytic continuation of the appropriate zeta functions is contructed and implemented. Among the results obtained are closed formulae for the individual determinants involved, the large $p$ behaviour of the determinants and the torsion, as well as an infinite set of distinct formulae for zeta(3): the ordinary Riemann zeta function evaluated at s=3. The torsion turns out to be trivial for the cases L(6,1), L((10,3) and L(12,5) and is, in general, greater than unity for large p and less than unity for a finite number of p and q.
-Riemann zeta function
Taekyun Kim
International Journal of Mathematics and Mathematical Sciences , 2004, DOI: 10.1155/s0161171204307180
Abstract: We consider the modified q-analogue of Riemann zeta function which is defined by ζq(s)=∑n=1∞(qn(s−1)/[n]s), 0
Riemann Zeta Function  [PDF]
Dorin Ghisa
Mathematics , 2009,
Abstract: Global mapping properties of the Riemann Zeta function are used to investigate its non trivial zeros.
Summability methods based on the Riemann Zeta function  [cached]
Larry K. Chu
International Journal of Mathematics and Mathematical Sciences , 1988, DOI: 10.1155/s0161171288000067
Abstract: This paper is a study of summability methods that are based on the Riemann Zeta function. A limitation theorem is proved which gives a necessary condition for a sequence x to be zeta summable. A zeta summability matrix Zt associated with a real sequence t is introduced; a necessary and sufficient condition on the sequence t such that Zt maps l1 to l1 is established. Results comparing the strength of the zeta method to that of well-known summability methods are also investigated.
Koshliakov kernel and identities involving the Riemann zeta function  [PDF]
Atul Dixit,Nicolas Robles,Arindam Roy,Alexandru Zaharescu
Mathematics , 2015,
Abstract: Some integral identities involving the Riemann zeta function and functions reciprocal in a kernel involving the Bessel functions $J_{z}(x), Y_{z}(x)$ and $K_{z}(x)$ are studied. Interesting special cases of these identities are derived, one of which is connected to a well-known transformation due to Ramanujan, and Guinand.
Zeros of the Riemann Zeta Function  [PDF]
Jin Gyu Lee
Mathematics , 2013,
Abstract: In this paper, we present a proof of the Riemann hypothesis. We show that zeros of the Riemann zeta function should be on the line with the real value 1/2, in the region where the real part of complex variable is between 0 and 1.
Identities for the Riemann zeta function  [PDF]
Michael O. Rubinstein
Mathematics , 2008,
Abstract: We obtain several expansions for $\zeta(s)$ involving a sequence of polynomials in $s$, denoted in this paper by $\alpha_k(s)$. These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities extend some series expansions for the zeta function that are known for integer values of $s$. The expansions also give a different approach to the analytic continuation of the Riemann zeta function.
Riemann zeta function is a fractal  [PDF]
S. C. Woon
Physics , 1994,
Abstract: Voronin's theorem on the `Universality'' of Riemann zeta function is shown to imply that Riemann zeta function is a fractal (in the sense that Mandelbrot set is a fractal) and a concrete ``representation'' of the ``giant book of theorems'' that Paul Halmos referred to.
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