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For q = 0 and q = t, one gets the Hall-Littlewood polynomial Pλ(x; t) and the Schur polynomial sλ, respectively, while limt→1 Pλ(x; t, t) yields the Jack polynomial P λ (x). Our first result is a generalization of Pλ(x;q, t) to n “t-parameters.” Thus let τ = (τ1, . . . , τn) be indeterminates, and put F = Q(q, τ). If μ is a partition, write q−μτ for the… (More)

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 [12] 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)

- Enrico Bombieri
- 2006

with s = 12 + it, and shows that ξ(t) is an even entire function of t whose zeros have imaginary part between −i/2 and i/2. He further states, sketching a proof, that in the range between 0 and T the function ξ(t) has about (T/2π) log(T/2π) − T/2π zeros. Riemann then continues “Man findet nun in der That etwa so viel reelle Wurzeln innerhalb dieser Grenzen,… (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)

n E ff e c t i v e r e s u l t s i n t h e d i o p h a n t i n e a p p r o x i m a t i o n o f a l g e b r a i c n u m b e r s a r e d i ffi c u l t t o o b t a i n , a n d f o r a l o n g t i m e t h e o n l y g e n e r a l m e t h o d a v a i l a b l e w a s B a k e r ’ s t h e o r y o f l i n e a r f o r m s i n l o g a r i t h m s . A n a l t e r n a t… (More)

- Enrico Bombieri, David C. Hunt, Alfred J. van der Poorten
- Experimental Mathematics
- 1995

Van der Poorten was supported in part by grants from the Australian Research Council and by a research agreement with Digital Equipment Corporation. 1991 Mathematics subject classi cation: 11J68, 11C20 Our investigations in the 1980’s of Thue’s method yielded determinants that we were only able to analyse successfully in part. We explain the context of our… (More)

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)

- Jeremy Gray, Andrew Wiles, +6 authors Landon T. Clay
- 2006

www.ams.org For additional information and updates on this book, visit www.ams.org/bookpages/mprize On August 8, 1900, at the second International Congress of Mathematicians in Paris, David Hilbert delivered his famous lecture in which he described twenty-three problems that were to play an infl uential role in mathematical research. A century later, on May… (More)