We give algorithms for the computation of the d-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log (2) or on a modest work station in a few hours run time. We demonstrate this technique by computing the ten billionth hexadecimal digit of, the billionth hexadecimal digits of 2 2 log(2) and log (2), and the ten billionth decimal digit of log(9=10). These calculations rest on the observation that very special types of identities exist for certain numbers like, 2,log(2) and log 2 (2). These are essentially polylogarithmic ladders in an integer base. A number of these identities that we deriveinthiswork appear to be new, for example the critical identity for:

: The asymptotic analysis of a class of binomial sums that arise in information theory can be performed in a simple way by means of singularity analysis of generating functions. The method developed extends the range of applicability of singularity analysis techniques to combinatorial sums involving transcendental elements like logarithms or fractional powers. Unit'e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T'el'ephone : (33) 01 39 63 55 11 -- T'el'ecopie : (33) 01 39 63 53 Analyse de singularit'e et asymptotique des sommes de Bernoulli R'esum'e : L'analyse asymptotique d'une classe de sommes qui interviennent en th'eorie de l'information peut etre effectu'ee de mani`ere simple par analyse de singularit'e de s'eries g'en'eratrices. La m'ethode d'evelopp'ee dans ce rapport 'etend en fait l'applicabilit'e des techniques d'analyse de singularit'e `a des sommes combinatoires comprenant des 'el'ements transcendants tels de...

In this paper a relationship between the Ohno relation for multiple zeta values and multiple polylogarithms are discussed. First we introduce generating functions for the Ohno relation, and investigate their properties. We show that there exists a subfamily of the Ohno relation which recovers algebraically its totality. This is proved through analysis of Mellin transform of multiple polylogarithms. Furthermore, this subfamily is shown to be converted to the Landen connection formula for multiple polylogarithms by inverse Mellin transform. 1

Cohen, Lewin and Zagier found four ladders that entail the polylogarithms

Abstract. We establish a series of integral formulae involving the Hurwitz zeta function. Applications are given to integrals of Bernoulli polynomials, log Γ(q) and log sin(q). 1.

There are known algorithms based on continued fractions for comparing fractions and for determining the sign of 2x2 determinants. The analysis of such extremely simple algorithms leads to an incursion into a surprising variety of domains. We take the reader through a light tour of dynamical systems (symbolic dynamics), number theory (continued fractions), special functions (multiple zeta values), functional analysis (transfer operators), numerical analysis (series acceleration), and complex analysis (the Riemann hypothesis). These domains all eventually contribute to a detailed characterization of the complexity of comparison and sorting algorithms, either on average or in probability.

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Abstract. A “hybrid method”, dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux’s method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions—this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.

Abstract. Buffon’s needle experiment is well-known: take a plane on which parallel lines at unit distance one from the next have been marked, throw a needle of unit length at random, and, finally, declare the experiment a success if the needle intersects one of the lines. Basic calculus implies that the probability of success is 2. π = 0.63661, and the experiment can be regarded as an analog (i.e., continuous) device that stochastically “computes ” 2 π. Generalizing the experiment and simplifying the computational framework, we ask ourselves which probability distributions can be produced perfectly, from a discrete source of unbiased coin flips. We describe and analyse a few simple Buffon machines that can generate geometric, Poisson, and logarithmic-series distributions (these are in particular required to transform continuous Boltzmann samplers of classical combinatorial structures into purely discrete random generators). Say that a number is Buffon if it is the probability of success of a probabilistic experiment based on discrete coin flippings. We provide human-accessible Buffon machines, which require a dozen coin flips or less, on average, and produce experiments whose probabilities are expressible in terms of numbers such as π, exp(−1), log 2, √ 3, cos ( 1 4), ζ(5). More generally, we develop a collection of constructions based on simple probabilistic mechanisms that enable one to create Buffon experiments involving compositions of exponentials and logarithms, polylogarithms, direct and inverse trigonometric functions, algebraic and hypergeometric functions, as well as functions defined by integrals, such as the Gaussian error function.

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