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

ISBN-10: 0817634819

ISBN-13: 9780817634810

ISBN-10: 1461234646

ISBN-13: 9781461234647

On April 25-27, 1989, over 100 mathematicians, together with 11 from overseas, accumulated 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 forthcoming (official) retirement of Paul T. Bateman, a trendy quantity theorist and member of the mathe matics school on the college 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 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 additional contributions. The papers span a variety of themes in quantity idea, with a majority in analytic quantity theory.

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MONTGOMERY S+(al,a2, ... , aR) denote the maximum number of sign changes with zero terms replaced by number of arbitrary sign. 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) .

Since we are supposing that R is large, we have = = = P(Z(sr) > 2/(sr -1/2)) = ~ 6, P(Z(sr) < -2/(sr -1/2)) ~ {, < r ~ R. Put Br = 1 if Z(sr) > 2/(sr -1/2), Br = -1 if Z(sr) < -2/(sr -1/2), and Br = 0 otherwise. Since the intervals (u(sr),v(sr)] are disjoint, the variables Z( sr) are independent. Hence Lemma 6 applies to for Rl the B r . Let PR = P(S-(BRl+1, BRl+2, ... , BR)) ~ {,(R - Rt}/5). By Lemma 6 we see that PR ~ exp (-6( R - Rd/3). For D E Q, 1/2 < s ~ 1, put UD(S) = -1, 0, or 1 according as KD(S) lies in (-00, -2/(s-I/2)), [-2/(s-I/2), 2/(s-I/2)], or (2/(s-I/2), +00), respectively.

5) 49 THE PRIME K-TUPLETS CONJECTURE uniformly for d $ 10gB x and (c, d) = 1. 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.

### Analytic Number Theory: Proceedings of a Conference in Honor of Paul T. Bateman by George E. Andrews (auth.), Bruce C. Berndt, Harold G. Diamond, Heini Halberstam, Adolf Hildebrand (eds.)

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