732 research outputs found
Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links
We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally
related to Dedekind zeta values, with coprime integers and giving for a manifold M
whose invariant trace field has a single complex place, discriminant ,
degree , and Dedekind zeta value . The largest numerator of the
998 invariants of Hodgson-Weeks manifolds is, astoundingly,
; the largest denominator is merely
b=9. We also study the rational invariant a/b for single-complex-place cusped
manifolds, complementary to knots and links, both within and beyond the
Hildebrand-Weeks census. Within the censi, we identify 152 distinct Dedekind
zetas rationally related to volumes. Moreover, 91 census manifolds have volumes
reducible to pairs of these zeta values. Motivated by studies of Feynman
diagrams, we find a 10-component 24-crossing link in the case n=2 and D=-20. It
is one of 5 alternating platonic links, the other 4 being quartic. For 8 of 10
quadratic fields distinguished by rational relations between Dedekind zeta
values and volumes of Feynman orthoschemes, we find corresponding links.
Feynman links with D=-39 and D=-84 are missing; we expect them to be as
beautiful as the 8 drawn here. Dedekind-zeta invariants are obtained for knots
from Feynman diagrams with up to 11 loops. We identify a sextic 18-crossing
positive Feynman knot whose rational invariant, a/b=26, is 390 times that of
the cubic 16-crossing non-alternating knot with maximal D_9 symmetry. Our
results are secure, numerically, yet appear very hard to prove by analysis.Comment: 53 pages, LaTe
Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k
Euler sums (also called Zagier sums) occur within the context of knot theory
and quantum field theory. There are various conjectures related to these sums
whose incompletion is a sign that both the mathematics and physics communities
do not yet completely understand the field. Here, we assemble results for
Euler/Zagier sums (also known as multidimensional zeta/harmonic sums) of
arbitrary depth, including sign alternations. Many of our results were obtained
empirically and are apparently new. By carefully compiling and examining a huge
data base of high precision numerical evaluations, we can claim with some
confidence that certain classes of results are exhaustive. While many proofs
are lacking, we have sketched derivations of all results that have so far been
proved.Comment: 19 pages, LaTe
Thirty-two Goldbach Variations
We give thirty-two diverse proofs of a small mathematical gem--the
fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also
discuss various generalizations for multiple harmonic (Euler) sums and some of
their many connections, thereby illustrating both the wide variety of
techniques fruitfully used to study such sums and the attraction of their
study.Comment: v1: 34 pages AMSLaTeX. v2: 41 pages AMSLaTeX. New introductory
material added and material on inequalities, Hilbert matrix and Witten zeta
functions. Errors in the second section on Complex Line Integrals are
corrected. To appear in International Journal of Number Theory. Title change
Phase transition in a log-normal Markov functional model
We derive the exact solution of a one-dimensional Markov functional model
with log-normally distributed interest rates in discrete time. The model is
shown to have two distinct limiting states, corresponding to small and
asymptotically large volatilities, respectively. These volatility regimes are
separated by a phase transition at some critical value of the volatility. We
investigate the conditions under which this phase transition occurs, and show
that it is related to the position of the zeros of an appropriately defined
generating function in the complex plane, in analogy with the Lee-Yang theory
of the phase transitions in condensed matter physics.Comment: 9 pages, 5 figures. v2: Added asymptotic expressions for the
convexity-adjusted Libors in the small and large volatility limits. v3: Added
one reference. Final version to appear in Journal of Mathematical Physic
An elementary proof of the irrationality of Tschakaloff series
We present a new proof of the irrationality of values of the series
in both qualitative and
quantitative forms. The proof is based on a hypergeometric construction of
rational approximations to .Comment: 5 pages, AMSTe
Expansion around half-integer values, binomial sums and inverse binomial sums
I consider the expansion of transcendental functions in a small parameter
around rational numbers. This includes in particular the expansion around
half-integer values. I present algorithms which are suitable for an
implementation within a symbolic computer algebra system. The method is an
extension of the technique of nested sums. The algorithms allow in addition the
evaluation of binomial sums, inverse binomial sums and generalizations thereof.Comment: 21 page
HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters
We present the Mathematica package HypExp which allows to expand
hypergeometric functions around integer parameters to arbitrary
order. At this, we apply two methods, the first one being based on an integral
representation, the second one on the nested sums approach. The expansion works
for both symbolic argument and unit argument. We also implemented new
classes of integrals that appear in the first method and that are, in part, yet
unknown to Mathematica.Comment: 33 pages, latex, 2 figures, the package can be downloaded from
http://krone.physik.unizh.ch/~maitreda/HypExp/, minor changes, works now
under Window
The Borwein brothers, Pi and the AGM
We consider some of Jonathan and Peter Borweins' contributions to the
high-precision computation of and the elementary functions, with
particular reference to their book "Pi and the AGM" (Wiley, 1987). Here "AGM"
is the arithmetic-geometric mean of Gauss and Legendre. Because the AGM
converges quadratically, it can be combined with fast multiplication algorithms
to give fast algorithms for the -bit computation of , and more
generally the elementary functions. These algorithms run in almost linear time
, where is the time for -bit multiplication. We
outline some of the results and algorithms given in Pi and the AGM, and present
some related (but new) results. In particular, we improve the published error
bounds for some quadratically and quartically convergent algorithms for ,
such as the Gauss-Legendre algorithm. We show that an iteration of the
Borwein-Borwein quartic algorithm for is equivalent to two iterations of
the Gauss-Legendre quadratic algorithm for , in the sense that they
produce exactly the same sequence of approximations to if performed using
exact arithmetic.Comment: 24 pages, 6 tables. Changed style file and reformatted algorithms in
v
Low energy expansion of the four-particle genus-one amplitude in type II superstring theory
A diagrammatic expansion of coefficients in the low-momentum expansion of the
genus-one four-particle amplitude in type II superstring theory is developed.
This is applied to determine coefficients up to order s^6R^4 (where s is a
Mandelstam invariant and R^4 the linearized super-curvature), and partial
results are obtained beyond that order. This involves integrating powers of the
scalar propagator on a toroidal world-sheet, as well as integrating over the
modulus of the torus. At any given order in s the coefficients of these terms
are given by rational numbers multiplying multiple zeta values (or
Euler--Zagier sums) that, up to the order studied here, reduce to products of
Riemann zeta values. We are careful to disentangle the analytic pieces from
logarithmic threshold terms, which involves a discussion of the conditions
imposed by unitarity. We further consider the compactification of the amplitude
on a circle of radius r, which results in a plethora of terms that are
power-behaved in r. These coefficients provide boundary `data' that must be
matched by any non-perturbative expression for the low-energy expansion of the
four-graviton amplitude.
The paper includes an appendix by Don Zagier.Comment: JHEP style. 6 eps figures. 50 page
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