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Random Matrix Theory and ζ(1/2 + it)
, 2000
"... We study the characteristic polynomials Z(U,#)of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of and Z/Z # , and from these we obtain the asymptotics of the value distributions and cumulants of the re ..."
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Cited by 85 (15 self)
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We study the characteristic polynomials Z(U,#)of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of and Z/Z # , and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N ##. In the
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
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Cited by 40 (5 self)
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Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.
Multiple Gamma Function and Its Application to Computation of Series, preprint
, 2003
"... Abstract. The multiple gamma function Γn, defined by a recurrencefunctional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma fu ..."
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Cited by 5 (1 self)
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Abstract. The multiple gamma function Γn, defined by a recurrencefunctional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the Γn function and their applications to summation of series and infinite products.
Contribution to the Theory of the Barnes Function
"... Abstract. This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for highprecision computation of the Barnes gamma function and Glaisher’s constant are also discussed. 1. ..."
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Cited by 4 (0 self)
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Abstract. This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for highprecision computation of the Barnes gamma function and Glaisher’s constant are also discussed. 1.
Symbolic and numeric computation of the Barnes functions
 2001 Electronic Proc. 7th Int. Conf. on Applications of Computer Algebra (Albuquerque, NM, 31 May–3
, 2001
"... This paper discusses some theoretical aspects and algorithms for highprecision computation of the Barnes gamma function. ..."
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Cited by 4 (0 self)
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This paper discusses some theoretical aspects and algorithms for highprecision computation of the Barnes gamma function.
1 1 rama.tex; 21/03/2011; 0:37; p.1Multiple Gamma Function and Its Application to Computation of Series and Products
"... Abstract. The multiple gamma function Γn, defined by a recurrencefunctional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the Barnes function h ..."
Abstract
 Add to MetaCart
Abstract. The multiple gamma function Γn, defined by a recurrencefunctional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the Barnes function has been revived. This paper discusses some theoretical aspects of the Γn function and their applications to summation of series and infinite products.