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, accrued on the collage of Illinois convention heart at Allerton Park for a big convention on analytic quantity thought. The occa­ sion marked the 70th birthday and approaching (official) retirement of Paul T. Bateman, a well known 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 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 additional contributions. The papers span a variety of themes in quantity conception, with a majority in analytic quantity theory.

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For any k-dimensional integer vector a, b or p we get a, b or p by omitting the last coordinate. Let x be a sufficiently large real number, and c and d be integers. e+b. d) We can now define our weight function W(a) by W(a)=l- L q:$;Q ~ q L ( L e(-fm)) e(am) m:$;qU (I,q)=l q = Lwme(am). 1) The parameters U and Q here are positive integers that will be chosen later, and the variable f runs over a reduced system of residues mod q. We start with the integral J = L dm~ d:$;D [1 ... 2) 52 ANTAL BALOG where D is any number satisfying 1 ~ D ~ x1/ 3 10g- B x.

Then S-(l;-oo,+oo) ~ S+(£;O,+oo). 313]. See also P6lya-Szego [18; vol 2, #33, 65,80]. OSCILLATIONS OF QUADRATIC L-FUNCTIONS 35 3. Proof of the Theorem Let DE Q. Taking a large value of R, we consider ~(s) at the points s Sr 1/2 + exp (_4r), RI < r ::; R where RI [R/5]. We take (J SR, A 12/«(J-l/2), and z (logX)A. We assume that X> XI(R), that X ::; IDI ::; 2X, and that D ¢ £«(J), so that the formula of Lemma 2 applies. Since A( n) = 0 if n is not a primepower, we may suppose that n pl:. The contribution of k ~ 3 in this formula is < 1, uniformly for S ~ 1/2.

000031671 ... 00003. 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.

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