We show that the \( abc\)-conjecture implies that few quadratic twists of a given
hyperelliptic curve have any non-trivial rational or integral points; and indicate how these
considerations dovetail with other predictions. Recently weak versions of these results have been proved unconditionally by Poonen and Stoll using methods first developed by Bhargava and Gross.Article

Analytic number theorists usually seek to show that sequences which appear
naturally in arithmetic are "well-distributed" in some appropriate sense.
In various discrepancy problems, combinatorics researchers have analyzed limitations
to equi-distribution, as have Fourier analysts when working with the
"uncertainty principle". In this article we find that these ideas have a natural
setting in the analysis of distributions of sequences in analytic number theory,
formulating a general principle, and giving several examples.Article

We give the first significant improvement to the Polya-Vinogradov theorem since 1919. This article highlights what was then the new concept of pretentiousness.Article

We discuss and reprove the Erdos-Turan theorem on the equidistribution in arguments of the roots of polynomials of small height, as well as Bilu's equidistribution theorem for algebraic numbers.Article

We present two proofs of Lang's conjecture for \( (G_m)^n\).Article

We give a relatively easy proof of the Erdos-Kac theorem via computing moments. We also
show how this proof extends naturally in a sieve theory context, and how it leads to several related
results in the literature.Article

An expository paper introducing the link between uniform distribution and exponential sumsArticle

We show that the set of prime factors of almost all integers are "Poisson distributed", and that this remains true (appropriately formulated) even when we restrict the number
of prime factors of the integer. Our results have inspired analogous results about the
distribution of cycle lengths of permutations. Article, and Corrigendum

We study the distribution function for \( |\zeta(1+it)|\), taking special care with the extreme values. Article

We discuss the relationship between various additive problems concerning
squares.Article

This is a slightly expanded write-up of my three lectures at the Additive Combinatorics
school. In the first lecture we introduce some of the basic material in Additive
Combinatorics, and in the next two lectures we prove two of the key background results, the
Freiman-Ruzsa theorem and Roth?s theorem for 3-term arithmetic progressions.Article

For fixed large \( x\) we give upper and lower bounds for the minimum
of \( \sum_{n\leq x} \chi(n)/n\) as we minimize over all real-valued Dirichlet characters \(\chi\).
Expanding our considerations to all multiplicative, real-valued
multiplicative functions of absolute value \( \leq 1\), the minimum equals
\( -0.4553 · · ·+o(1)\), and in this case we can classify the set of optimal functions.Article

We present three remarks on Goldbach's problem. First we suggest a refinement of
Hardy and Littlewood's conjecture for the number of representations of \( 2n\) as the sum of two
primes, positing an estimate with a very small error term. Next we show that if a strong form
of Goldbach's conjecture is true then every even integer is the sum of two primes from a rather
sparse set of primes. Finally we show that an averaged strong form of Goldbach's conjecture is
equivalent to the Generalized Riemann Hypothesis; as well as a similar equivalence to estimates
for the number of ways of writing integers as the sum of \( k\) primes.Article