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 overseas, collected on the collage of Illinois convention heart at Allerton Park for an enormous convention on analytic quantity conception. The occa­ sion marked the 70th birthday and drawing close (official) retirement of Paul T. Bateman, a admired 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 math division. The convention featured a complete of fifty-four talks, together with ten in­ vited lectures through 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 themes in quantity conception, with a majority in analytic quantity theory.

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Hence <:: Xy l - 2v (log X)2H8 <:: X N l - 2v (log N)2k. DeQ IDI~X By the Cauchy-Schwarz inequality we see that IT212 <:: [00 (log y)2k y-V-1/2 dy Here the first integral is follows that [00 ISD(y)1 2y-v-3/2 dy. <:: y1/2-V(logy)2k(q -12)-1. Hence by (9) it OSCILLATIONS OF QUADRATIC L-FUNCTIONS 31 On combining these estimates, we find that L 1L2 12 ~ X N 1- 2t1 (log N)2k. DEQ IDI$X On the other hand, from Lemma 1 we deduce that L L IL112 ~ X DEQ (logm)k(logn)k(mn)-u. mn=D N

8] A. Smati, Repartition des valeurs de la fonction d' Euler, L' Ens. Math. 35 (1989), 61-76. Michel Balazard and Abdelhakim Smati Departement de Mathematiques Faculte des Sciences 123 Av. A. Thomas 87060 Limoges Cedex France The Prime k-Tuplets Conjecture on Average ANTAL BALOG Dedicated to Professor Paul Bateman on the occasion of his 70th birthday 1. Introduction. The well-known twin prime conjecture states that there are infinitely many primes p such that p + 2 is also a prime. Although the proof of this seemingly simple statement is hopeless at present many further connected conjectures exist.

Let x be a real number with x ~ Ih I and let Z = Z(bl:; x) be the set of k-dimensional integer vectors b such that the last coordiante of b is bl: and the set {n : 1 < an + b ::; x} is not empty. 8) The implied constant in the symbol ~ depends at most on A and a but not on bl:. This will be important later. In other words, if we fix a and the last coordinate of b then we can prove the prime k-tuplets conjecture on 50 ANTAL BALOG average over the other coordinates of b. In addition we can do this in the form of a strong distribution theorem in residue classes.

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