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By Gordon M. Barrow

Creation to Molecular Spectroscopy
Gordon M. Barrow

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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 [8] proved(l) under the assumption that 12AI :::: 31AI-4 using Freiman's structure theory of set addition (see [4]). As noted by Nathanson and Jia [6], (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).

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