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 another country, accrued on the collage of Illinois convention heart at Allerton Park for an immense convention on analytic quantity thought. The occa sion marked the 70th birthday and coming near near (official) retirement of Paul T. Bateman, a popular 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 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 extra contributions. The papers span quite a lot of subject matters in quantity conception, with a majority in analytic quantity theory.

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**Example text**

In his proof of N (x) '" Ax, he approximated the Euler function by the truncated function

3 log x , then 3logx N(x, y) ~ N(x) ~ N(x(1- --)-1, y). y (3) Proof: The first inequality follows from

Math. 31 (1979), 476-487. [17] G. P6lya, Verschiedene Bemerkung zur Zahlentheorie, Jber. Deutsch. Math. Verein 28 (1919), 31-40. [18] G. P6lya and G. Szego, Aufgaben und Lehrsiitze aus der Analysis, Vols I and II, Springer-Verlag (Berlin), 1964. [19] D. Wolke, Eine Bemerkung liber das Legendre-Symbol, Monat. Math. 77 (1973), 267-275. R. C. Baker Royal Holloway and Bedford New College Egham Surrey TW 20 OEX England, U. K. Hugh L. Montgomery Department of Mathematics University of Michigan Ann Arbor, MI 48109-1003 USA Elementary Proof of a Theorem of Bateman MICHEL BALAZARD AND ABDELHAKIM SMATI Dedicated to Paul T.

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 . logz peR) = hmsup --NS(R)n'P(Z) x-oo Z then lim peR) = 0. R--+oo With more work we could replace Sr by a sequence tending to 1/2 more slowly.