Download Analytic Number Theory: Proceedings of a Conference In Honor by Krishnaswami Alladi (auth.), Bruce C. Berndt, Harold G. PDF

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

On may possibly sixteen -20, 1995, nearly one hundred fifty mathematicians amassed on the convention heart of the college of Illinois at Allerton Park for an Inter­ nationwide convention on Analytic quantity conception. The assembly marked the impending professional retirement of Heini Halberstam from the math fac­ ulty of the college of Illinois at Urbana-Champaign. Professor Halberstam has been on the collage due to the fact 1980, for eight years as head of the dep. 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 through 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 contain contributions from many of the important audio system and from a number of the different individuals, in addition to a few papers from mathematicians who have been not able to wait. The contents span a huge variety of subject matters from modern quantity thought, with the bulk having an analytic flavor.

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Extra resources for Analytic Number Theory: Proceedings of a Conference In Honor of Heini Halberstam Volume 1

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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.

Let ~ 1. {an}~=l be an SSD-sequence. Then 32 lAEGUG BAE Note that all the elements of J are positive integers. Since A has the SSDproperty, B, B' Hence III we have C oF B' A and B = 2n -1. Because a1 + a2 + ... + an is the greatest element in J, As {I, 2, 22 ,2 3 " .. } suggests, SSD-sequences are quite sparse. It seems very natural to ask how dense they can be. We will consider a question of this flavor in this paper. As a way of obtaining finite "dense" SSD-sets, one can use the Conway-Guy sequence (see [8], [10]).

S < "'2(i-l)" = ___ ~ • - ~ 1-2s for all positive integers m and all real numbers s Proof. 2, for all positive integers k ~ ~ o. 1) for non-negative s. 3) m ' " a. s m > '" 2(i-l)s ~ • - ~ i=l i=l 1 2ms = _-_ _ 1-2" for all positive integers m and all s ~ O? 3) is true for s = 1. What about other positive values of s? In connection with this question we have the following theorem. 4. Let {an}:=l be an SSD-sequence and f3 a fixed positive real number. fJ > ~ 2(i-l),8 • - ~ i=l i=l ~ for all positive integers m, and all 0 ~ s ~ f3 .

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