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

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 a foreign country, collected on the collage of Illinois convention middle at Allerton Park for an important convention on analytic quantity thought. The occa­ sion marked the 70th birthday and forthcoming (official) retirement of Paul T. Bateman, a favourite quantity theorist and member of the mathe­ matics college on the college 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 additional contributions. The papers span a variety of subject matters in quantity conception, with a majority in analytic quantity theory.

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

L: cp(a)~:c/y (4) 1 L: cp(a)~:c l~b~:c/cp(a) The estimation of these sums depends on the following three lemmas. Lemma 2. Suppose y and k are real numbers so that y ~logy. Put u 1- k/logy. Then = L: tp( a )-U 0, we have - = L: p-l + L:(pl-U - P~ y P~ Y 1)p-l p~y ~ L: p-l + (eA: - 1)(logy)-1 L: p-1logp p~y p~y ~ loglogy+ O(eA:), and the Lemma follows.

Such ad E 1) induces <: X1/2U- 1/ 2 discriminants DE Q with IDI ~ X. Re(8) ~ 1, 19 Xo(IT). Suppose that D E Q \ C(IT) so that LD(8) #; 0 for 8 E 'R1(IT) and IT - 1/2 ~ l/log IDI .

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