1,184 research outputs found
On Lerch's transcendent and the Gaussian random walk
Let be independent variables, each having a normal distribution
with negative mean and variance 1. We consider the partial sums
, with , and refer to the process as
the Gaussian random walk. We present explicit expressions for the mean and
variance of the maximum These expressions are in terms
of Taylor series about with coefficients that involve the Riemann
zeta function. Our results extend Kingman's first-order approximation [Proc.
Symp. on Congestion Theory (1965) 137--169] of the mean for .
We build upon the work of Chang and Peres [Ann. Probab. 25 (1997) 787--802],
and use Bateman's formulas on Lerch's transcendent and Euler--Maclaurin
summation as key ingredients.Comment: Published at http://dx.doi.org/10.1214/105051606000000781 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Huyghens, Bohr, Riemann and Galois: Phase-Locking
Several mathematical views of phase-locking are developed. The classical
Huyghens approach is generalized to include all harmonic and subharmonic
resonances and is found to be connected to 1/f noise and prime number theory.
Two types of quantum phase-locking operators are defined, one acting on the
rational numbers, the other on the elements of a Galois field. In both cases we
analyse in detail the phase properties and find them related respectively to
the Riemann zeta function and to incomplete Gauss sums.Comment: 18 pages paper written in relation to the ICSSUR'05 conference held
in Besancon, France to be published at a special issue of IJMP
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