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Research Abstracts Online
January 2008 - March 2009

University of Minnesota Twin Cities
Institute of Technology
School of Mathematics

PI: Andrew Odlyzko

Statistical Analyses of Values of the Riemann Zeta Function

This project investigates the idea that the asymptotic moments of the zeta function can be modeled by the asymptotic moments of characteristic polynomials of unitary random matrices. The researchers are also testing whether the value distributions of the zeta function at finite heights can be modeled by the value distributions of characteristic polynomials of finite matrices in the unitary ensemble.

The researchers have applied their methods to about 20 billion zeros near zero 10[super]23 as well as smaller sets of zeros at lower heights. So far, moments of zeta up to the 12th moment have been calculated. In addition, extreme value behavior, derivatives, and value distribution of zeta have been investigated. The validity of the resulting data has been confirmed by comparing it to recent zeta computations of Rubinstein and by repeating calculations on overlapping sets.

The results indicate that, in general, numerics seem to converge to Random Matrix Theory predictions. But there is still a significant amount of variation in the data for high moments (particularly sixth and higher). This is in stark contrast to gap distributions of zeros, which exhibit much faster convergence. Recent moment conjectures by Conrey et al., which go beyond the leading asymptotic term predicted by RMT, have also been tested. Although data show a remarkable degree of agreement with their conjectures at low heights, this ceases to be the case when one reaches heights as low as 10[super]10. The researchers continue to analyze the data.

Group Members

Ghaith Hiary, Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada
Vijay Kumar Adhikari, Graduate Student