The Riemann Zeros and Eigenvalue Asymptotics

The Riemann Zeros and Eigenvalue Asymptotics (1999)

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by M. V. Berry , J. P. Keating

Venue:	SIAM Rev
Citations:	39 - 4 self

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Datum	Value	Source
TITLE	The Riemann Zeros and Eigenvalue Asymptotics	user correction - Legacy Corrections
AUTHOR NAME	M. V. Berry	SVM HeaderParse 0.1
AUTHOR NAME	J. P. Keating	SVM HeaderParse 0.1
ABSTRACT	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 random-matrix 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 Riemann-Siegel 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.	user correction - Legacy Corrections
YEAR	1999	INFERENCE
VENUE	SIAM Rev	INFERENCE
VENUE TYPE	JOURNAL	INFERENCE
PAGES	236--266	INFERENCE
VOLUME	41	INFERENCE
CITATIONS	70 found	ParsCit 1.0