23 research outputs found
P\'olya-Vinogradov and the least quadratic nonresidue
It is well-known that cancellation in short character sums (e.g. Burgess'
estimates) yields bounds on the least quadratic nonresidue. Scant progress has
been made on short character sums since Burgess' work, so it is desirable to
find a new approach to nonresidues. The goal of this note is to demonstrate a
new line of attack via long character sums, a currently active area of
research. Among other results, we demonstrate that improving the constant in
the P\'{o}lya-Vinogradov inequality would lead to significant progress on
nonresidues. Moreover, conditionally on a conjecture on long character sums, we
show that the least nonresidue for any odd primitive character (mod ) is
bounded by .Comment: 9 pages; a few small corrections from the previous versio
The distribution of the maximum of character sums
We obtain explicit bounds on the moments of character sums, refining
estimates of Montgomery and Vaughan. As an application we obtain results on the
distribution of the maximal magnitude of character sums normalized by the
square root of the modulus, finding almost double exponential decay in the tail
of this distribution.Comment: 16 pages, 1 figure, new version with correction
Lower bounds on odd order character sums
A classical result of Paley shows that there are infinitely many quadratic
characters whose character sums get as large as ; this implies that a conditional upper bound of Montgomery and Vaughan
cannot be improved. In this paper, we derive analogous lower bounds on
character sums for characters of odd order, which are best possible in view of
the corresponding conditional upper bounds recently obtained by the first
author.Comment: 6 page
L-functions with n-th order twists
Let K be a number field containing the n-th roots of unity for some n > 2. We
prove a uniform subconvexity result for a family of double Dirichlet series
built out of central values of Hecke L-functions of n-th order characters of K.
The main new ingredient, possibly of independent interest, is a large sieve for
n-th order characters. As further applications of this tool, we derive several
results concerning L(s,\chi) for n-th order Hecke characters: an estimate of
the second moment on the critical line, a non-vanishing result at the central
point, and a zero-density theorem.Comment: 21 pages, 1 figur
On the spectral distribution of large weighted random regular graphs
McKay proved that the limiting spectral measures of the ensembles of
-regular graphs with vertices converge to Kesten's measure as
. In this paper we explore the case of weighted graphs. More
precisely, given a large -regular graph we assign random weights, drawn from
some distribution , to its edges. We study the relationship
between and the associated limiting spectral distribution
obtained by averaging over the weighted graphs. Among other results, we
establish the existence of a unique `eigendistribution', i.e., a weight
distribution such that the associated limiting spectral
distribution is a rescaling of . Initial investigations suggested
that the eigendistribution was the semi-circle distribution, which by Wigner's
Law is the limiting spectral measure for real symmetric matrices. We prove this
is not the case, though the deviation between the eigendistribution and the
semi-circular density is small (the first seven moments agree, and the
difference in each higher moment is ). Our analysis uses
combinatorial results about closed acyclic walks in large trees, which may be
of independent interest.Comment: Version 1.0, 19 page
Character sums to smooth moduli are small
Recently, Granville and Soundararajan have made fundamental breakthroughs in
the study of character sums. Building on their work and using estimates on
short character sums developed by Graham-Ringrose and Iwaniec, we improve the
Polya-Vinogradov inequality for characters with smooth conductor.Comment: 18 pages. Section 5 significantly revise