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Journals and Conferences
In a recent paper Xian-Jin Li showed that the Riemann Hypothesis holds if and only if n = P 1?(1?1==) n has n > 0 for n = 1; 2; 3; : : : where runs over the complex zeros of the Riemann zeta function. We show that Li's criterion follows as a consequence of a general set of inequalities for an arbitrary multiset of complex numbers and therefore is not… (More)
E. Bombieri  has written at length concerning applications of the large sieve to number theory. Our intent here is to complement his exposition by devoting our attention to the analytic principle of the large sieve; we describe only briefly how applications to number theory are made. The large sieve was studied intensively during the decade 1965-1975,… (More)
When a fixed algebraic variety in a multiplicative group variety is intersected with the union of all algebraic subgroups of fixed dimension, a key role is played by what we call the anomalous subvarieties. These arise when the algebraic variety meets translates of subgroups in sets larger than expected. We prove a Structure Theorem for the anomalous… (More)
1 Voltaire said that Archimedes had more imagination than Homer. Unfortunately, as far as we know, Archimedes's only use of it, outside of mathematics, was to condemn goldsmiths and to kill people. More peaceful uses of mathematicians' imagination, to the outside of mathematics, in decreasing order of impact, were provided by Multi-Millionaire Richard… (More)
I. Let Q(N; q; a) denote the number of squares in the arithmetic progression qn+a; n = 1; 2; ; N; and let Q(N) be the maximum of Q(N; q; a) over all non-trivial arithmetic progressions qn + a. It seems to be remarkably diicult to obtain non-trivial upper bounds for Q(N). There are currently two proofs known of the weak bound Q(N) = o(N) (which is an old… (More)
Our investigations in the 1980s of Thue's method yielded determinants which we were only able to analyse successfully in part. We explain the context of our work, recount our experiences, mention our conjectures, and allude to a number of open questions.
The Riemann zeta function is the function of the complex variable s, defined in the half-plane 1 (s) > 1 by the absolutely convergent series ζ(s) := ∞ n=1 1 n s , and in the whole complex plane C by analytic continuation. As shown by Riemann, ζ(s) extends to C as a meromorphic function with only a simple pole at s = 1, with residue 1, and satisfies the… (More)
We present in a unified way proofs of Roth's theorem and an effective version of Mordell's conjecture, using the ABC conjecture. We also show how certain stronger forms of the ABC conjecture give information about the type of approximation to an algebraic number.
We show that if r > n!(n!-2) the set of solutions x, E C(t) of a Fern-rat equation 1; u,x; = 0, a, E C(t), is the union of at most n! " ! families with an explicitly given simple structure. In particular, the number of projective solutions, up to rth roots of unity, of such an equation is either at most n! " ' or infinite. The proof uses the function held… (More)