By Krishnaswami Alladi (auth.), Bruce C. Berndt, Harold G. Diamond, Adolf J. Hildebrand (eds.)

On may well sixteen -20, 1995, nearly a hundred and fifty mathematicians accumulated on the convention middle of the collage of Illinois at Allerton Park for an Inter nationwide convention on Analytic quantity concept. The assembly marked the impending authentic retirement of Heini Halberstam from the maths fac ulty of the collage of Illinois at Urbana-Champaign. Professor Halberstam has been on the college considering the fact that 1980, for eight years as head of the dept of arithmetic, and has been a number one researcher and instructor in quantity concept for over 40 years. this system integrated invited one hour lectures by way of G. Andrews, J. Bour achieve, J. M. Deshouillers, H. Halberstam, D. R. Heath-Brown, H. Iwaniec, H. L. Montgomery, R. Murty, C. Pomerance, and R. C. Vaughan, and nearly 100 different talks of various lengths. those volumes include contributions from lots of the significant audio system and from the various different members, in addition to a few papers from mathematicians who have been not able to wait. The contents span a extensive diversity of subject matters from modern quantity concept, with the bulk having an analytic flavor.

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A convention on Analytic quantity thought and Diophantine difficulties was once held from June 24 to July three, 1984 on the Oklahoma kingdom collage in Stillwater. The convention used to be funded through the nationwide technology starting place, the school of Arts and Sciences and the dept of arithmetic at Oklahoma kingdom collage.

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6), sm(q) is the generating function for partitions of the type enumerated by B (n) with the added condition that each part is ~ m - 1. 7) Therefore Theorem 6 is the q-analog of Theorem 4. 11). In Section 2 we assemble the necessary background. In Section 3 we prove Theorem 5 and in Section 4 we prove Theorem 6. Section 5 is devoted to the limits of the reciprocal polynomials q2N 2 -2N T2N( q-l), q2N 2 T2N+l (q-l), q2N 2 -2 S2N (q-l), and q2N 2 +2N S2N+l(q-l). We conclude with a discussion of open questions.

Suppose that A, 17, II satisfy (A 2 ), (A 3 ), and A satisfies (Al) as well. 7) Here 8 0 depends on 8. (K, L, M, Q) «: xC-e. Proof. This follows from [4], Theorem 3. Lemma 5. Let KLM = x, min(K,L,M) > x'1, II = (Ilk)' k '" K, A = (At), l '" L, 17 = (17m ), m '" M. Suppose that II, A, 17 satisfy (A 2 ), (A3) and A satisfies (Al) as well. Let 8 > o. 10) Proof. This is a slight variant of Theorem 4 of [4]. 10). It is remarked in [4] that the condition on K could be relaxed, but the improvement is not stated there.

Th. 10 (1971), 266-270. ___ , The Theory of Partitions, Encyclopedia of Math. , Vol. -C. , Addison-Wesley, Reading, 1976 (Reprinted: Cambridge University Press, London, 1985). ___ , Euler's "Exemplum memorabile inductionis fallacis" and qtrinomial coefficients, J. Amer. Math. Soc. 3 (1990), 653-669. G. E. Andrews and R. J. Baxter, Lattice gas generalizations of the hard hexagon model, Ill. q-trinomial coefficients, J. of Stat. Physics 47 (1987), 297-330. N. J. Fine, Basic Hypergeometric Series and Applications, Math Surveys and Monographs, No.