By George E. Andrews (auth.), Bruce C. Berndt, Harold G. Diamond, Heini Halberstam, Adolf Hildebrand (eds.)

On April 25-27, 1989, over 100 mathematicians, together with 11 from in a foreign country, accrued on the collage of Illinois convention heart at Allerton Park for a big convention on analytic quantity concept. The occa sion marked the 70th birthday and approaching (official) retirement of Paul T. Bateman, a fashionable quantity theorist and member of the mathe matics school on the collage of Illinois for nearly 40 years. For fifteen of those years, he served as head of the maths division. The convention featured a complete of fifty-four talks, together with ten in vited lectures by way of H. Delange, P. Erdos, H. Iwaniec, M. Knopp, M. Mendes France, H. L. Montgomery, C. Pomerance, W. Schmidt, H. Stark, and R. C. Vaughan. This quantity represents the contents of thirty of those talks in addition to extra contributions. The papers span a variety of issues in quantity conception, with a majority in analytic quantity theory.

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6) below) 7r 7r{x; (I, 1), (O, h); c, d) and T T{x; (I, 1), (O, h)). In the same paper he announced a generalization of this result to prime k-tuplets. The proof . has appeared in [6]. 6) for any A > 0, where 8 > 0 is some small computable constant. 6) to improve the size of the largest known gap between consecutive primes. 6) coupled with a lower bound sieve to deduce that infinitely often there are three primes and an almost prime in arithmetic progression. This was originally proved by HeathBrown [4] in 1981.

Obviously p(p) p(p) or p(p) p(p) - 1, and p(p) p implies p(p) p. But p(p) p can also happen when p(p) < p. Thus 1I'(x; a, h; c, d) 0 is possible when O'(a, b; g, q) # O. This must be also reflected in the "singular series" . In section 6 we will evaluate this "singular series" by proving the following recursion formula. = = = = = = = Lemma 3. 5), (n) is the squarefree part of n (that is, the product of the distinct prime divisors ofn), v(n) is the number of distinct prime divisors of n, and Ivl is the length of the vector v.

Otherwise put 1/2 + exp (_4r), put AD(S) = °. For r = 1,2, ... NS(R)(Z), x-oo Z Then lim q(R) = 0. R--+oo Similarly, if . logz peR) = hmsup --NS(R)n'P(Z) x-oo Z then lim peR) = 0. R--+oo With more work we could replace Sr by a sequence tending to 1/2 more slowly. We note several consequences of our main result. °. Corollary 1. Let K be a given number. The set of D E Q for which S-(LiJ; 1/2, 1) ::; K has asymptotic density It seems likely that S-(LiJ; 1/2,1) ::::: log log IDI for most D. For DE Q let IDI (D) FD(z) = ~ -; zN be the associated Fekete polynomial.