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32
On the Random Character of Fundamental Constant Expansions
 EXPERIMENTAL MATHEMATICS
, 2001
"... We propose a theory to explain random behavior for the digits
in the expansions of fundamental mathematical constants. At
the core of our approach is a general hypothesis concerning the distribution of the iterates generated by dynamical maps. On this main hypothesis, one obtains proofs of base2 no ..."
Abstract

Cited by 55 (16 self)
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We propose a theory to explain random behavior for the digits
in the expansions of fundamental mathematical constants. At
the core of our approach is a general hypothesis concerning the distribution of the iterates generated by dynamical maps. On this main hypothesis, one obtains proofs of base2 normality—namely bit randomness in a specific technical sense— for a collection of celebrated constants, including , log 2, (3), and others. Also on the hypothesis, the number (5) is either rational or normal to base 2. We indicate a research connection between our dynamical model and the theory of pseudorandom number
generators.
Random Generators and Normal Numbers
 EXPERIMENTAL MATHEMATICS
, 2000
"... Pursuant to the authors' previous chaoticdynamical model for random digits of fundamental constants [3], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results such as the following are achieved: Whereas the fundamental ..."
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Cited by 25 (11 self)
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Pursuant to the authors' previous chaoticdynamical model for random digits of fundamental constants [3], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results such as the following are achieved: Whereas the fundamental constant log 2 = P n2Z + 1=(n2 n ) is not yet known to be 2normal (i.e. normal to base 2), we are able to establish bnormality (and transcendency) for constants of the form P 1=(nb n ) but with the index n constrained to run over certain subsets of Z + . In this way we demonstrate, for example, that the constant 2;3 = P n=3;3 2 ;3 3 ;::: 1=(n2 n ) is 2normal. The constants share with ; log 2 and others the property that isolated digits can be directly calculated, but for the new class such computation is extraordinarily rapid. For example, we find that the googolth (i.e. 10 100  th) binary bit of 2;3 is 0. We also present a collection of other results  such as density results and irrationality proofs based on PRNG ideas  for various special numbers.
Central Binomial Sums and Multiple Clausen Values (with Connections to Zeta Values
"... We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of nonalternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of al ..."
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Cited by 22 (9 self)
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We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of nonalternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. In the nonalternating case, there is a strong connection to polylogarithms of the sixth root of unity, encountered in the 3loop Feynman diagrams of hepth/9803091 and subsequently in hepph/9910223, hepph/9910224, condmat/9911452 and hepth/0004010.
Fast Multiplication And Its Applications
"... This survey explains how some useful arithmetic operations can be sped up from quadratic time to essentially linear time. ..."
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Cited by 20 (4 self)
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This survey explains how some useful arithmetic operations can be sped up from quadratic time to essentially linear time.
Lattice sums for the Helmholtz equation
 SIAM Review
"... Abstract. A survey of different representations for lattice sums for the Helmholtz equation is given. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension d of the underlying space and ..."
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Cited by 11 (2 self)
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Abstract. A survey of different representations for lattice sums for the Helmholtz equation is given. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension dΛ. Lattice sums are related to, and can be calculated from, the quasiperiodic Green’s function and this object serves as the starting point of the analysis.
Digital Sums And DivideAndConquer Recurrences: Fourier Expansions And Absolute Convergence
, 2004
"... We propose means of computing the Fourier expansions of periodic functions appearing in higher moments of the sumofdigits function and in the solutions of some divideandconquer recurrences. The expansions are shown to be absolutely convergent. We also give a new approach to efficiently computing ..."
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Cited by 7 (2 self)
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We propose means of computing the Fourier expansions of periodic functions appearing in higher moments of the sumofdigits function and in the solutions of some divideandconquer recurrences. The expansions are shown to be absolutely convergent. We also give a new approach to efficiently computing numerically the coefficients involved to high precision.
Computation and theory of extended MordellTornheimWitten sums
 Mathematics of Computation
, 2013
"... Abstract. We consider some fundamental generalized Mordell–Tornheim–Witten (MTW) zetafunction values along with their derivatives, and explore connections with multiplezeta values (MZVs). To achieve this, we make use of symbolic integration, high precision numerical integration, and some interestin ..."
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Cited by 5 (4 self)
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Abstract. We consider some fundamental generalized Mordell–Tornheim–Witten (MTW) zetafunction values along with their derivatives, and explore connections with multiplezeta values (MZVs). To achieve this, we make use of symbolic integration, high precision numerical integration, and some interesting combinatorics and specialfunction theory. Our original motivation was to represent unresolved constructs such as Eulerian loggamma integrals. We are able to resolve all such integrals in terms of a MTW basis. We also present, for a substantial subset of MTW values, explicit closedform expressions. In the process, we significantly extend methods for highprecision numerical computation of polylogarithms and their derivatives with respect to order.
Spaceefficient evaluation of hypergeometric series
 SIGSAM Bulletin, Communications in Computer Algebra
"... Many important constants, such as e and Apéry’s constant ζ(3), can be approximated by a truncated hypergeometric series. The evaluation of such series to high precision has traditionally been done by binary splitting followed by fixedpoint division. However, the numerator and the denominator comput ..."
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Cited by 4 (4 self)
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Many important constants, such as e and Apéry’s constant ζ(3), can be approximated by a truncated hypergeometric series. The evaluation of such series to high precision has traditionally been done by binary splitting followed by fixedpoint division. However, the numerator and the denominator computed by binary splitting usually contain a very large common factor. In this paper, we apply standard computer algebra techniques including modular computation and rational reconstruction to overcome the shortcomings of the binary splitting method. The space complexity of our algorithm is the same as a bound on the size of the reduced numerator and denominator of the series approximation. Moreover, if the predicted bound is small, the time complexity is better than the standard binary splitting approach. Our approach allows a series to be evaluated to a higher precision without additional memory. We show that when our algorithm is applied to compute ζ(3), the memory requirement is significantly reduced compared to the binary splitting approach. 1