Analytic Number Theory: Proceedings of a Conference in Honor by George E. Andrews (auth.), Bruce C. Berndt, Harold G.

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 out of the country, amassed on the collage of Illinois convention middle at Allerton Park for an incredible convention on analytic quantity idea. The occa­ sion marked the 70th birthday and coming near near (official) retirement of Paul T. Bateman, a well-liked 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 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 additional contributions. The papers span a variety of subject matters in quantity conception, with a majority in analytic quantity theory.

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Also, 1 - 0 ~ e- 6 • Hence MK ~ (10e)K+l e -6R ~ (10e)6R/5 e-6R. But (10e)1/5 ~ e 2/ 3 , so we have the desired bound. It is not hard to show that and that Thus when oR is large the number of sign changes is usually"" oR. Indeed, the distribution function of S- resembles that of a binomial variable with parameters R, O. Hence it is not surprising that should be small. We conclude this section by quoting a lemma of real analysis which we use in deriving Corollary 2. In its simplest form, this lemma is Descartes' rule of signs.

ZR are independent random variables such that P(Zr > 0) ~ 0 and P(Zr < 0) ~ 0 for all r. Then 1 P(S-(Zl, Z2, ... , ZR) ~ 50R) ~ e- 6R /3 uniformly in 0 and R. Proof: First we reduce to a simpler situation. Without loss of generality we may assume that our probability space is [0,1]R , and that Zr(x) is a function of Xr alone, where x (Xl, X2, ... , XR). For each r let Yr Yr(x r ) be defined so that if Yr < 0 then Zr < 0, if Yr > 0 then Zr > 0, and with the further property that P(Yr < 0) P(Yr > 0) 0 .

To state this result we have to generalize our notation, and let 1I"(x; a, b; c, d) be the number of integers n in the residue class c (mod d) such that 1 < an + b :::; x and an + b are primes. 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.

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