This work presents a class of methods by which one can translate, on a term-by-term basis, an asymptotic expansion of a function around a dominant singularity into a corresponding asymptotic expansion for the Taylor coefficients ofthe function. This approach is based on contour integration using Cauchy's formula and Hankel-like contours. It constitutes an alternative to either Darboux's method or Tauberian theorems that appears to be well suited to combinatorial enumerations, and a few applications in this area are outlined. Key words, asymptotic analysis, generating functions, combinatorial enumeration AMS(MOS) subject classifications. 05, 40, 68 1. Introduction. Several applications in analysis, especially combinatorics, necessitate determining the asymptotic order of growth of coefficients of a function that is analytic at the origin. It has been recognized for a long time that the function's dominant singularities (the ones of smallest modulus) contain a great deal of information on the coefficients. This paper describes a very general method based on earlier works of ours (Odlyzko [1982], Flajolet and Odlyzko [1982 ]) that applies to functions of "moderate" variation. We restrict our attention to functions with a unique dominant singularity. (Functions with a finite number of singularities on their circle of convergence can also be treated by a direct extension of our methods, using composite integration contours.) By normalization, we may always assume that the dominant singularity occurs at z l, and we consider functions that satisfy, for some arbitrary real number a, f(z),(l-z) (z-1). Letf be a sequence of numbers, with a generating function f(z) Zn_0 fz that is analytic at the origin. In nonelementary cases that we encounter in combinatorial analysis, the f are only accessible via f(z), and f(z) itself is either explicitly defined by a closed-form expression or implicitly specified as the solution to a functional equation. (See, for instance, Comtet [1974], Goulden and Jackson [1983 ], Stanley [1986 ].) The problem is thus to obtain estimates forf from whatever analytic information is available onf(z). For example, the generating function of 2-regular graphs (Comtet [1974 ]) is f(z) c-z/2 z2/4 and the expansion off(z) at z 1, (1.1a) f(z)=e-3/4[ f_z+ l-z+ ],

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A numerical study of the distribution of spacings between zeros of the Riemann zeta function is presented. It is based on values for the first 105 zeros and for zeros number 1012 + 1 to 1012 + 105 that are accurate to within +10"8, and which were calculated on the Cray-1 and Cray X-MP computers. This study tests the Montgomery pair correlation conjecture as well as some further conjectures that predict that the zeros of the zeta function behave like eigenvalues of random Hermitian matrices. Matrices of this type are used in modeling energy levels in physics, and many statistical properties of their eigenvalues are known. The agreement between actual statistics for zeros of the zeta function and conjectured results is generally good, and improves at larger heights. Several initially unexpected phenomena were found in the data and some were explained by relating them to the primes.

Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.

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