Results 1 
3 of
3
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 ..."
Abstract

Cited by 52 (10 self)
 Add to MetaCart
(Show Context)
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.
SUMMARY
"... This paper describes an extension to ditroff/ffortid, a system for formatting bidirectional text in Arabic, Hebrew, and Persian. The previous version of the system is able to format mixed lefttoright and righttoleft text using fonts with separated letters or with connecting letters and only con ..."
Abstract
 Add to MetaCart
This paper describes an extension to ditroff/ffortid, a system for formatting bidirectional text in Arabic, Hebrew, and Persian. The previous version of the system is able to format mixed lefttoright and righttoleft text using fonts with separated letters or with connecting letters and only connection stretching, achieved by repeating fixed lenth baseline fillers. The latest extension adds the abilities to stretch letters themselves, as is common in Arabic, Hebrew, and Persian calligraphic printing, and to slant the baselines of words, as is common in Persian calligraphic printing. The extension consists of modifications in ffortid that allow it to interface with (1) dynamic POSTSCRIPT fonts to which one can pass to the outline procedure for any stretchable and/or connected letter, parameters specifying the amounts of stretch for the letter itself and/or for the connecting parts of the letter and (2) POSTSCRIPT fonts whose characters are slanted so that merely applying show to a word ends up printing that entire word on a single slanted baseline. As a selftest, this paper was formatted using the described system, and it contains many
Contents
"... Abstract. In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted Lfunctions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest ..."
Abstract
 Add to MetaCart
Abstract. In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted Lfunctions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number theory, such as lower order terms in the zero statistics of Lfunctions, mollified moments of Lfunctions and discrete averages over zeros of the Riemann zeta function. In particular, using the ratios conjectures we easily derive the answers to a number of notoriously difficult