277 research outputs found
Fourier Expansion of the Riemann zeta function and applications
We study the distribution of values of the Riemann zeta function
on vertical lines , by using the theory of Hilbert space.
We show among other things, that, has a Fourier expansion in the
half-plane and its Fourier coefficients are the binomial
transform involving the Stieltjes constants. As an application, we show
explicit computation of the Poisson integral associated with the logarithm of
. Moreover, we discuss our results with respect to the
Riemann and Lindel\"{o}f hypotheses on the growth of the Fourier coefficients.Comment: 21 page
The Chang-Refsdal Lens Revisited
This paper provides a complete theoretical treatment of the point-mass lens
perturbed by constant external shear, often called the Chang-Refsdal lens. We
show that simple invariants exist for the products of the (complex) positions
of the four images, as well as moment sums of their signed magnifications. The
image topographies and equations of the caustics and critical curves are also
studied. We derive the fully analytic expressions for precaustics, which are
the loci of non-critical points that map to the caustics under the lens
mapping. They constitute boundaries of the region in the image domain that maps
onto the interior of the caustics. The areas under the critical curves,
caustics and precaustics are all evaluated, which enables us to calculate the
mean magnification of the source within the caustics. Additionally, the exact
analytic expression for the magnification distribution for the source in the
triangular caustics is derived, as well as a useful approximate expression.
Finally, we find that the Chang-Refsdal lens with the convergence greater than
unity can exhibit third-order critical behaviour, if the reduced shear is
exactly equal to \sqrt{3}/2, and that the number of images for N-point masses
with non-zero constant shear cannot be greater than 5N-1.Comment: to appear in MNRAS (including 6 figures, 3 appendices; v2 - minor
update with corrected typos etc.
One-Loop BPS amplitudes as BPS-state sums
Recently, we introduced a new procedure for computing a class of one-loop
BPS-saturated amplitudes in String Theory, which expresses them as a sum of
one-loop contributions of all perturbative BPS states in a manifestly T-duality
invariant fashion. In this paper, we extend this procedure to all BPS-saturated
amplitudes of the form \int_F \Gamma_{d+k,d} {\Phi}, with {\Phi} being a weak
(almost) holomorphic modular form of weight -k/2. We use the fact that any such
{\Phi} can be expressed as a linear combination of certain absolutely
convergent Poincar\'e series, against which the fundamental domain F can be
unfolded. The resulting BPS-state sum neatly exhibits the singularities of the
amplitude at points of gauge symmetry enhancement, in a chamber-independent
fashion. We illustrate our method with concrete examples of interest in
heterotic string compactifications.Comment: 42 pages; v4: a few misprints correcte
Lower bounds for discrete negative moments of the Riemann zeta function
We prove lower bounds for the discrete negative th moment of the derivative of the Riemann zeta function for all fractional . The bounds are in line with a conjecture of Gonek and Hejhal. Along the way, we prove a general formula for the discrete twisted second moment of the Riemann zeta function. This agrees with a conjecture of Conrey and Snaith. <br
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