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

On might sixteen -20, 1995, nearly one hundred fifty mathematicians amassed on the convention heart of the collage of Illinois at Allerton Park for an Inter nationwide convention on Analytic quantity idea. The assembly marked the impending respectable retirement of Heini Halberstam from the maths fac ulty of the college of Illinois at Urbana-Champaign. Professor Halberstam has been on the collage on account that 1980, for eight years as head of the dept of arithmetic, and has been a number one researcher and instructor in quantity thought 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 virtually 100 different talks of various lengths. those volumes contain contributions from lots of the critical audio system and from some of the different members, in addition to a few papers from mathematicians who have been not able to wait. The contents span a extensive diversity of topics from modern quantity conception, with the bulk having an analytic flavor.

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**Example text**

Given B(n) in Theorem 1, (Ll) where Iql < 1. In a subsequent paper [3], further results related to these theorems were derived. Of particular interest for us here is the following result which follows immediately from the work in [3]: Theorem 3. 2) 1 Partially supported by National Science Foundation Grant DMS 8702695-04. 18 GEORGE E. ANDREWS where C(n) is the number of partitions of n which neither contain sequences nor allow any part to appear more than twice. In 1990, in a seemingly completely unrelated paper [6], an extensive study of trinomial and q-trinomial coefficients was given.

A. 35 (1949), 408-411. edu The Brun-Titchmarsh Theorem on average R. C. Bakerl and G. Harman To Professor Heini Halberstam on the occasion of his retirement 1. Introduction Throughout this paper a denotes a fixed non-zero integer and the letter p with or without subscript denotes a prime variable. As usual, for (q, a) = 1 we write L 7r(x;q,a) = 1. p:5x p=a(mod q) By the form of the Brun-Titchmarsh inequality given by Montgomery and Vaughan [24] we know that 2x 7r(x; q, a) < ifJ(q) log(x/q)' for 1 ::; q < x.

5) XO. 676 for some E > 0 and X ~ EX $p$x+a xo(a, E). Now, writing A(n) for the Von Mangoldt THE BRUN-TITCHMARSH THEOREM ON AVERAGE 43 function, L L A(t) = L L logp2 + O(x). Llog(p+a) = Pl~" P21Pl +a Assuming x is large, it follows that L (log p)1r(Xj p, -a) = L L logp2 > (1- f)X. P1~" P21p1 +a It is an easy consequence of the Bombieri-Vinogradov theorem [6] and the Brun-Titchmarsh theorem in simplest form [24] that for x ~ xo(a, f). 5). 4985x as required. This completes the deduction of Theorem 2.