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, accumulated on the college of Illinois convention heart at Allerton Park for an important convention on analytic quantity idea. The occa sion marked the 70th birthday and drawing close (official) retirement of Paul T. Bateman, a well known quantity theorist and member of the mathe matics college on the college 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 via 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 additional contributions. The papers span a variety of subject matters in quantity thought, with a majority in analytic quantity theory.

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**Extra resources for Analytic Number Theory: Proceedings of a Conference in Honor of Paul T. Bateman**

**Sample text**

Thus for example, S-(l,O,l) = while S+(l,O, 1) = 2. In any case, S- ::; S+ . If! (a2), ... (aR» over all finite sequences for which a < al < a2 < ... < aR < b . These conventions are standard (see Karlin [15]). We show that most LD(S) are very far from being monotonic in the interval (1/2,1). ° Theorem. Suppose that AD(S) = L' S > 1/2. If 1~(s)1 > l/(s - 1/2) then set ~(s) . 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 .

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. The conjecture in the title, for example, asks if k linear polynomials with suitable conditions on the coefficients represent simultaneously primes infinitely often.

841 ... 159.... Further examination reveals that Z(s) depends most heavily on the primes in the vicinity of exp (l/(s - 1/2)) . J p p' _ p-' = U(I)