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

Analytic

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

ISBN-10: 146128645X

ISBN-13: 9781461286455

On may perhaps 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 thought. The assembly marked the impending reputable retirement of Heini Halberstam from the maths fac­ ulty of the college of Illinois at Urbana-Champaign. Professor Halberstam has been on the collage in view that 1980, for eight years as head of the dep. of arithmetic, and has been a number one researcher and instructor in quantity conception 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 virtually 100 different talks of various lengths. those volumes include contributions from many of the critical audio system and from a number 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 subject matters from modern quantity conception, with the bulk having an analytic flavor.

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3. A comment on a lower bound for an :=1 Let {an} be an SSD-sequence. 59-60]' [8] and [12]). This problem can be stated inversely: Under the condition an ~ x, find an upper bound on n. In his paper [3], N. Elkies mentioned the inter-relation between a lower bound on an and an upper bound on n in terms of x. 2) 1 n ~ log2 x + S log2 10g2 X + log2 C + 0(1) . 114, problem C8]). [ii by an analytic method. 2) immediately from the following theorem of Moser. 1. For any integer n ~ n E ai i=l 1, we have n 2 > E 2 (i-1) .

14) is proved, 5. The reciprocal polynomials In various applications of results like Theorems 5 and 6, it is useful to know the limiting behavior of the reciprocal polynomials, Theorem 7. (5,1) lim q2N 2-2N r2N (q-1) = N--+oo 1 ~ q12j2+5j (1 (q2; q2)00 ,~ + q6J+2) )=-00 (_q7; q24)00( _q17; q24)00(q24; q24)00 + q2( _q; q24)00( _q23; q24)00(q24; q24)00 (q2; q2)00 28 GEORGE E. 4) Proof. 6) . 7) n-l (A; q)n = II (1 - Aqj). 1). ~ q )=-00 2. \ = (q , q)oo = = 2. 1). The remainder follow in the same way.

I) In the next section, we consider the Dirichlet series of an SSD-sequence, namely, :'=1 m L a;" for real s. i=l (ii) In the third section, we consider a lower bound on L. Moser asserts that m L i=l m ai 2 > L2 2 (i-1) . i=l We give a comment regarding this theorem. an. Here a result of ON SUBSET-SUM-DISTINCT SEQUENCES 33 2. Tomic's Inequality First of all, we introduce TomiC's inequality. It will turn out that his result m is extremely useful for the estimation of La/, especially when s is not an i=l integer.

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