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.

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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.

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

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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.

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

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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.

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

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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.

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

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We discuss the relationship between various additive problems concerning squares.

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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.

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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.

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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.

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