By Nicholas M. Katz

The most subject of this booklet is the deep relation among the spacings among zeros of zeta and $L$-functions and spacings among eigenvalues of random parts of huge compact classical teams. This relation, the Montgomery-Odlyzko legislation, is proven to carry for huge periods of zeta and $L$-functions over finite fields. The ebook attracts on and provides obtainable money owed of many disparate parts of arithmetic, from algebraic geometry, moduli areas, monodromy, equidistribution, and the Weil conjectures, to chance conception at the compact classical teams within the restrict as their size is going to infinity and comparable ideas from orthogonal polynomials and Fredholm determinants.

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4. Fix a real number a. } to (a, a + 27r) N (ordered). (2) For each integer k > 1, denote by Wk the locally closed set in PN,s' consisting of those f which have a zero at T = eia of exact multiplicity k. This bijection is continuous on Wk. PROOF. It suffices to treat the case a = 0. (1) Any element f in PN,St-{1} lies in PN,Z[E,2a-E] for some c > 0. If f --+ f is a Cauchy sequence in PN,SI, we claim that each fn, itself lies in PN,Z[E/2,2a-E/2] as soon as n >> 0. To see this, take any point a in Z[-e/2,,-/2]- Then we have dist(a, Z[e, 27r - E]) > e/7r (the circular distance is > e/2), and hence our f in PN,Z[e,2a-E] satisfies If (a) I ?

Tr, E Clump(a)(s)Ta = a>O Sep(b)(s)T6 = b>O Sep(b)(s)(1 + T)b, b>O C1ump(a)(s)(T - 1)a. a>O PROOF. The first identity is, coefficient by coefficient, the identity of the lemma. The second identity is obtained from the first by the change of variable T - T-1. QED 2. 11. For each a > 0 in Z', we have the identity Sep(a)(s) _ (-1)"-LBinom(n, a) Clump(n)(s). 12. For each a > 0 in Zr, and each integer m > E(a), we have the inequalities (-1)n-a Binom(n, a) Clump(n)(s) < Sep(a)(s) a

A, H(N), offsets c), p,(univ, offsets c))d Haar JH(N) discrep(µ(A, G(N), offsets c), p(univ, offsets c))dHaar. G(N) 1. STATEMENTS OF THE MAIN RESULTS 28 In case d), we have f discrep(u(A, H(N), offsets c), u(univ, offsets c))d Haar H(N) = either discrep(g(A, SO(2N), offsets c), u(univ, offsets c))dHaar fSO(2N) or f discrep(p,(A, O(2N), offsets c), u(univ, offsets c))dHaar, (2N) depending on whether or not H(N) acts by inner automorphism on SO(2N). 6 extended). 6 holds. Let N be an integer > 1, G(N) C H(N) compact groups in one of the following four cases: 'ti a) G(N) = SU(N) C H(N) C normalizer of G(N) in U(N), b) G(N) = SO(2N + 1) C H(N) c normalizer of G(N) in U(2N + 1), c) G(N) = USp(2N) C H(N) C normalizer of G(N) in U(2N), d) G(N) = SO(2N) C H(N) C normalizer of G(N) in U(2N).