The main topic of this book is the deep relation between the spacings between zeros of zeta and $L$-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and $L$-functions over finite fields. The book draws on and gives accessible accounts of many disparate areas of mathematics, from algebraic geometry, moduli spaces, monodromy, equidistribution, and the Weil conjectures, to probability theory on the compact classical groups in the limit as their dimension goes to infinity and related techniques from orthogonal polynomials and Fredholm determinants.
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"[F]or research workers interested in the Riemann Hypothesis, or in the arithmetic of varieties over finite fields, this work has important messages which may help to shape our thinking on fundamental issues on the nature of zeta-functions." ---- Bulletin of the London Mathematical Society
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Inside This Book
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First Sentence:
In a remarkable numerical experiment, Odlyzko [Odl] found that the distribution of the (suitably normalized) spacings between successive zeroes of the Riemann zeta function is (empirically) the same as the so-called GUE measure, a certain probability measure on R arising in random matrix theory. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
inequality discrep, geometric monodromy group, corresponding separation vector, proper smooth family, irreducible nontrivial representation, double limit formulas, compact classical groups, cardinalities increase, axiomatic situation, polarized abelian scheme, geometric fibre, spacing measures, nonvoid open set, spectral determinants, equidistribution theorem, normalized angles, spacing vectors, total mass one, abelian schemes, finite total mass, asserted estimate, tail estimate, normalized spacings, connected fibres, principally polarized abelian varieties Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proj Aut, Intrin Card, R-valued Borel, Ofer Gabber, Key Lemma, Suppose Theorem, Hermann Weyl, Main Theorem, Taking the Haar New!
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Front Cover | Table of Contents | First Pages | Back Cover | Surprise Me!
In a remarkable numerical experiment, Odlyzko [Odl] found that the distribution of the (suitably normalized) spacings between successive zeroes of the Riemann zeta function is (empirically) the same as the so-called GUE measure, a certain probability measure on R arising in random matrix theory. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
inequality discrep, geometric monodromy group, corresponding separation vector, proper smooth family, irreducible nontrivial representation, double limit formulas, compact classical groups, cardinalities increase, axiomatic situation, polarized abelian scheme, geometric fibre, spacing measures, nonvoid open set, spectral determinants, equidistribution theorem, normalized angles, spacing vectors, total mass one, abelian schemes, finite total mass, asserted estimate, tail estimate, normalized spacings, connected fibres, principally polarized abelian varieties Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proj Aut, Intrin Card, R-valued Borel, Ofer Gabber, Key Lemma, Suppose Theorem, Hermann Weyl, Main Theorem, Taking the Haar New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Back Cover | Surprise Me!
Citations (learn more)
This book cites 2
books:
- Algebraic Geometry: The Johns Hopkins Centennial Lectures by Professor Jun-Ichi Igusa in Back Matter
- Algebraic Geometry (Dover Books on Mathematics) by Solomon Lefschetz in Back Matter
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