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Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem
In this paper we study generalised prime systems for which the integer counting function NP(x) is asymptotically well behaved, in the sense that NP(x)=ρx+O(xβ), where ρ is a positive constant and . For such systems, the associated zeta function ζP(s) is holomorphic for . We prove that for , for any ε>0, and also for ε=0 for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in NkP(x)−Ress=1(ζPk(s)xs/s), which is O(xθ) for some θ<1. Letting αk denote the infimum of such θ, we show that
Estimates for compositions of maximal operators with singular integrals
We prove weak-type (1,1) estimates for compositions of maximal operators with
singular integrals. Our main object of interest is the operator
where is Bourgain's maximal multiplier operator and is the
sum of several modulated singular integrals; here our method yields a
significantly improved bound for the operator norm when . We
also consider associated variation-norm estimates
Myth and reality : points of departure in American literature and culture in the nineteenth century
On Differences of Zeta Values
Finite differences of values of the Riemann zeta function at the integers are
explored. Such quantities, which occur as coefficients in Newton series
representations, have surfaced in works of Maslanka, Coffey, Baez-Duarte, Voros
and others. We apply the theory of Norlund-Rice integrals in conjunction with
the saddle point method and derive precise asymptotic estimates. The method
extends to Dirichlet L-functions and our estimates appear to be partly related
to earlier investigations surrounding Li's criterion for the Riemann
hypothesis.Comment: 18 page
Connectedness properties of the set where the iterates of an entire function are bounded
We investigate some connectedness properties of the set of points K(f) where
the iterates of an entire function f are bounded. In particular, we describe a
class of transcendental entire functions for which an analogue of the
Branner-Hubbard conjecture holds and show that, for such functions, if K(f) is
disconnected then it has uncountably many components. We give examples to show
that K(f) can be totally disconnected, and we use quasiconformal surgery to
construct a function for which K(f) has a component with empty interior that is
not a singleton.Comment: 21 page
Field Quantization in 5D Space-Time with Z-parity and Position/Momentum Propagator
Field quantization in 5D flat and warped space-times with Z-parity is
comparatively examined. We carefully and closely derive 5D
position/momentum(P/M) propagators. Their characteristic behaviours depend on
the 4D (real world) momentum in relation to the boundary parameter () and
the bulk curvature (\om). They also depend on whether the 4D momentum is
space-like or time-like. Their behaviours are graphically presented and the
Z symmetry, the "brane" formation and the singularities are examined. It is
shown that the use of absolute functions is important for properly treating the
singular behaviour. The extra coordinate appears as a {\it directed} one like
the temperature. The problem, which is an important consistency
check of the bulk-boundary system, is solved {\it without} the use of
KK-expansion. The relation between P/M propagator (a closed expression which
takes into account {\it all} KK-modes) and the KK-expansion-series propagator
is clarified. In this process of comparison, two views on the extra space
naturally come up: orbifold picture and interval (boundary) picture.
Sturm-Liouville expansion (a generalized Fourier expansion) is essential there.
Both 5D flat and warped quantum systems are formulated by the Dirac's bra and
ket vector formalism, which shows the warped model can be regarded as a {\it
deformation} of the flat one with the {\it deformation parameter} \om. We
examine the meaning of the position-dependent cut-off proposed by
Randall-Schwartz.Comment: 44 figures, 22(fig.)+41 pages, to be published in Phys.Rev.D, Fig.4
is improve
Antichaos in a Class of Random Boolean Cellular Automata
A variant of Kauffman's model of cellular metabolism is presented. It is a
randomly generated network of boolean gates, identical to Kauffman's except for
a small bias in favor of boolean gates that depend on at most one input. The
bias is asymptotic to 0 as the number of gates increases. Upper bounds on the
time until the network reaches a state cycle and the size of the state cycle,
as functions of the number of gates , are derived. If the bias approaches 0
slowly enough, the state cycles will be smaller than for some . This
lends support to Kauffman's claim that in his version of random network the
average size of the state cycles is approximately .Comment: 12 pages. A uuencoded, tar-compressed postscipt file containing
figures has been adde
Deep-inelastic scattering and the operator product expansion in lattice QCD
We discuss the determination of deep-inelastic hadron structure in lattice
QCD. By using a fictitious heavy quark, direct calculations of the Compton
scattering tensor can be performed in Euclidean space that allow the extraction
of the moments of structure functions. This overcomes issues of operator mixing
and renormalisation that have so far prohibited lattice computations of higher
moments. This approach is especially suitable for the study of the twist-two
contributions to isovector quark distributions, which is practical with current
computing resources. While we focus on the isovector unpolarised distribution,
our method is equally applicable to other quark distributions and to
generalised parton distributions. By looking at matrix elements such as
(where and are vector
and axial-vector heavy-light currents) within the same formalism, moments of
meson distribution amplitudes can also be extracted.Comment: 10 pages, 5 figures, version accepted for publicatio
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