Lillian B. Pierce

ON A DISCRETE VERSION OF TANAKA'S THEOREM FOR MAXIMAL FUNCTIONS

In this paper we prove a discrete version of Tanakas Theorem (19) for the Hardy-Littlewood maximal operator in dimension n = 1, both in the non-centered and centered cases. For the non-centered maximal operator f we prove that, given a function f : Z → R of bounded variation, Var(f Mf) ≤ Var(f), where Var(f) represents the total variation of f. For the centered maximal operator M we prove that, given a function f : Z → R such that f ∈ l 1 (Z), Var(Mf) ≤ Ckfkl1(Z).

The 3-part of Class Numbers of Quadratic Fields

It is proved that the 3-part of the class number of a quadratic field is in general and if has a divisor of size . These bounds follow as results of nontrivial estimates for the number of solutions to the congruence modulo in the ranges and , where are nonzero integers and is a square-free positive integer. Furthermore, it is shown that the number of elliptic curves over with conductor is in general and if has a divisor of size . These results are the first improvements to the trivial bound and the resulting bound for the 3-part and the number of elliptic curves, respectively.

Discrete fractional Radon transforms and quadratic forms

We consider discrete analogues of fractional Radon transforms involving integration over paraboloids defined by positive definite quadratic forms. We prove that such discrete operators extend to bounded operators from

A bound for the 3-part of class numbers of quadratic fields by means of the square sieve

\ell^p

Burgess bounds for short mixed character sums

to

Lower bounds for the truncated Hilbert transform

\ell^q

On discrete fractional integral operators and mean values of Weyl sums

for a certain family of kernels. The method involves an intricate spectral decomposition according to major and minor arcs, motivated by ideas from the circle method of Hardy and Littlewood. Techniques from harmonic analysis, in particular Fourier transform methods and oscillatory integrals, as well as the number theoretic structure of quadratic forms, exponential sums, and theta functions, play key roles in the proof.

Averages and moments associated to class numbers of imaginary quadratic fields

Abstract We prove a nontrivial bound of O(|D|27/56+ε) for the 3-part of the class number of a quadratic field ℚ(√D) by using a variant of the square sieve and the q-analogue of van der Corputs method to count the number of squares of the form 4x 3 − dz 2 for a square-free positive integer d and bounded x, z.

Polynomial Carleson Operators Along Monomial Curves in the Plane

This paper proves nontrivial bounds for short mixed character sums by introducing estimates for Vinogradovs mean value theorem into a version of the Burgess method.

Women’s History Month

Given two intervals