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

Analytic

By George E. Andrews (auth.), Bruce C. Berndt, Harold G. Diamond, Heini Halberstam, Adolf Hildebrand (eds.)

ISBN-10: 0817634819

ISBN-13: 9780817634810

On April 25-27, 1989, over 100 mathematicians, together with 11 from in another country, accumulated on the college of Illinois convention middle at Allerton Park for an immense convention on analytic quantity idea. The occa­ sion marked the 70th birthday and forthcoming (official) retirement of Paul T. Bateman, a in demand 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 math 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 extra contributions. The papers span quite a lot of issues in quantity conception, with a majority in analytic quantity theory.

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Extra resources for Analytic Number Theory: Proceedings of a Conference in Honor of Paul T. Bateman

Example text

First, we observe that pslI (1 + p(p~ 1») = A + 0(;) . Then, if z is large enough, y > 4 log z and logy> Uog(z(1- 31;g~)-1)P/2 , Lemmas 1 and 5 give TI N(z) = A z + O(z y- 1 Iogz) + Of(Z logy e-(1-f)ulog u). We choose y = eJt log~ loglog~ , so that ~1----­ u log u = (1 + 0(1)) 2log z log log z, and this completes the proof of (1). In conclusion, let us point out that the prime number theorem is not required in our proof (neither in Bateman's proof, by the way). This is in contrast to [6] and [8], where the prime number theorem with remainder term is an essential tool.

Any improvement on Lemma 2 automatically improves this exponent (up to 1/2). One can consider Vinogradov's celebrated result [9] as the special case D 1 of Lemma 2. In fact our proof follows very closely Vaughan's proof [8] for Vinogradov's estimate. As we have mentioned earlier we defer the proof of these lemmata to later sections, and we now return to the study of the integral J here. 9) By the Dirichlet approximation theorem we can find for any 0' a rational number u/v such that z (u,v)=I, l~v~Q' u Q 10'--1<-.

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. Bateman 1. The distribution of values of the Euler totient function

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