By Krishnaswami Alladi (auth.), K. Alladi, P. D. T. A. Elliott, A. Granville, G. Tenebaum (eds.)
This quantity features a choice of papers in Analytic and trouble-free quantity conception in reminiscence of Professor Paul Erdös, one of many maximum mathematicians of this century. Written by means of many best researchers, the papers care for the latest advances in a large choice of issues, together with arithmetical services, best numbers, the Riemann zeta functionality, probabilistic quantity thought, homes of integer sequences, modular varieties, walls, and q-series.
Audience: Researchers and scholars of quantity idea, research, combinatorics and modular kinds will locate this quantity to be stimulating.
Read Online or Download Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdös PDF
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Extra resources for Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdös
THE RAMANUJAN JOURNAL 2, 39-45 (1998) © 1998 Kluwer Academic Publishers. Manufactured in The Netherlands. yu Katedra Matematike RGF-a, Universitet u Beogradu, Dusina 7, 11000 Beograd, Serbia, Yugoslavia Dedicated to the memory of Paul Erdos, who proved and conjectured more than anyone else Received January 16, 1997; Accepted June 2, 1997 Abstract. The classical Voronoi identity is proved in a relatively simple way by the use of the Laplace transform. Here b (x) denotes the error term in the Dirichlet divisor problem, d(n) is the number of divisors of nand K 1, Y1 are the Bessel functions.
Thus, by (2), IEh(A)I:::: ~(lhAI + IAhl):;:: ~(12AI + IA 21):::: ~IE2(A)I. (4) We also have In particular, if (1) fails for a particular h, it fails for all larger h. When h = 2, (1) has been established for certain very special sets of positive integers A. Nathanson and Tenenbaum  proved(l) under the assumption that 12AI :::: 31AI-4 using Freiman's structure theory of set addition (see ). As noted by Nathanson and Jia , (1) can also be proved in the case where A is contained in a "short" interval of length 1Aio(log21Ail using the fact that log d(n) = O(log njlog 2 n), where d(n) is the number of divisors of n.
If 1 = L[f(x)] 00 f(x)e-sx dx is the (one-sided) Laplace transform of f(x), then for ffies > 0 L[~(x)] = = foo Jo (L'd(n)- x(logx n::'Ox ~ loo n=l n ~d(n) sx e- dx + 2y- 1)- ~) e-sx dx + logs-y 2 1 -- 4s S 1~ =- ~d(n)e s n=l -sn + logs - y 1 1 ~ (w)f(w)s-w s2 1 -4s 1 = --. ~ 2 (w)f(w)s-w dw 2JTls c2) y + logs2 2 2_, = - 1-. 2JT 1s 012) dw- s 1 s - -4 (3) 4s Here we used the well-known Mellin integral e-z = - 1-. 2JTI r lee) f(w)w-z dw (c > 0, ffiez > 0), and the series representation ~ 2 (s) = 00 Ld(n)n-s (ffies > 1).