9 research outputs found
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
This paper sketches a technique for improving the rate of convergence of a
general oscillatory sequence, and then applies this series acceleration
algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may
be taken as an extension of the techniques given by Borwein's "An efficient
algorithm for computing the Riemann zeta function", to more general series. The
algorithm provides a rapid means of evaluating Li_s(z) for general values of
complex s and the region of complex z values given by |z^2/(z-1)|<4.
Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an
Euler-Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in
that two evaluations of the one can be used to obtain a value of the other;
thus, either algorithm can be used to evaluate either function. The
Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta,
while the Borwein algorithm is superior for evaluating the polylogarithm in the
kidney-shaped region. Both algorithms are superior to the simple Taylor's
series or direct summation.
The primary, concrete result of this paper is an algorithm allows the
exploration of the Hurwitz zeta in the critical strip, where fast algorithms
are otherwise unavailable. A discussion of the monodromy group of the
polylogarithm is included.Comment: 37 pages, 6 graphs, 14 full-color phase plots. v3: Added discussion
of a fast Hurwitz algorithm; expanded development of the monodromy
v4:Correction and clarifiction of monodrom
The distinct flavors of Zipf's law in the rank-size and in the size-distribution representations, and its maximum-likelihood fitting
In the last years, researchers have realized the difficulties of fitting
power-law distributions properly. These difficulties are higher in Zipf's
systems, due to the discreteness of the variables and to the existence of two
representations for these systems, i.e., two versions about which is the random
variable to fit. The discreteness implies that a power law in one of the
representations is not a power law in the other, and vice versa. We generate
synthetic power laws in both representations and apply a state-of-the-art
fitting method (based on maximum-likelihood plus a goodness-of-fit test) for
each of the two random variables. It is important to stress that the method
does not fit the whole distribution, but the tail, understood as the part of a
distribution above a cut-off that separates non-power-law behavior from
power-law behavior. We find that, no matter which random variable is power-law
distributed, the rank-size representation is not adequate for fitting, whereas
the representation in terms of the distribution of sizes leads to the recovery
of the simulated exponents, may be with some bias
Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli
The Dirichlet characters of reduced residue systems modulo m are tabulated
for moduli m <= 195. The associated L-series are tabulated for m <= 14 and
small positive integer argument s accurate to 10^(-50), their first derivatives
for m <= 6. Restricted summation over primes only defines Dirichlet Prime
L-functions which lead to Euler products (Prime Zeta Modulo functions). Both
are materialized over similar ranges of moduli and arguments. Formulas and
numerical techniques are well known; the aim is to provide direct access to
reference values.Comment: Table extended to cover moduli <= 195 in the ancillary director
Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models
We address the nonperturbative structure of topological strings and c=1
matrix models, focusing on understanding the nature of instanton effects
alongside with exploring their relation to the large-order behavior of the 1/N
expansion. We consider the Gaussian, Penner and Chern-Simons matrix models,
together with their holographic duals, the c=1 minimal string at self-dual
radius and topological string theory on the resolved conifold. We employ Borel
analysis to obtain the exact all-loop multi-instanton corrections to the free
energies of the aforementioned models, and show that the leading poles in the
Borel plane control the large-order behavior of perturbation theory. We
understand the nonperturbative effects in terms of the Schwinger effect and
provide a semiclassical picture in terms of eigenvalue tunneling between
critical points of the multi-sheeted matrix model effective potentials. In
particular, we relate instantons to Stokes phenomena via a hyperasymptotic
analysis, providing a smoothing of the nonperturbative ambiguity. Our
predictions for the multi-instanton expansions are confirmed within the
trans-series set-up, which in the double-scaling limit describes
nonperturbative corrections to the Toda equation. Finally, we provide a
spacetime realization of our nonperturbative corrections in terms of toric
D-brane instantons which, in the double-scaling limit, precisely match
D-instanton contributions to c=1 minimal strings.Comment: 71 pages, 14 figures, JHEP3.cls; v2: added refs, minor change
Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions
We present a systematic short time expansion for the generating function of
the one point height probability distribution for the KPZ equation with droplet
initial condition, which goes much beyond previous studies. The expansion is
checked against a numerical evaluation of the known exact Fredholm determinant
expression. We also obtain the next order term for the Brownian initial
condition. Although initially devised for short time, a resummation of the
series allows to obtain also the \textit{long time large deviation function},
found to agree with previous works using completely different techniques.
Unexpected similarities with stationary large deviations of TASEP with periodic
and open boundaries are discussed. Two additional applications are given. (i)
Our method is generalized to study the linear statistics of the {Airy point
process}, i.e. of the GUE edge eigenvalues. We obtain the generating function
of the cumulants of the empirical measure to a high order. The second cumulant
is found to match the result in the bulk obtained from the Gaussian free field
by Borodin and Ferrari, but we obtain systematic corrections to the Gaussian
free field (higher cumulants, expansion towards the edge). This also extends a
result of Basor and Widom to a much higher order. We obtain {large deviation
functions} for the {Airy point process} for a variety of linear statistics test
functions. (ii) We obtain results for the \textit{counting statistics of
trapped fermions} at the edge of the Fermi gas in both the high and the low
temperature limits.Comment: 84 pages, 3 figure