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 base-2 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.

Pursuant to the authors' previous chaotic-dynamical 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 2-normal (i.e. normal to base 2), we are able to establish b-normality (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 2-normal. 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 googol-th (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.

This survey explains how some useful arithmetic operations can be sped up from quadratic time to essentially linear time.

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 non-alternating 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 non-alternating case, there is a strong connection to polylogarithms of the sixth root of unity, encountered in the 3-loop Feynman diagrams of hep-th/9803091 and subsequently in hep-ph/9910223, hep-ph/9910224, cond-mat/9911452 and hep-th/0004010.

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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 fixed-point 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

The flow of the Riemann zeta function, ˙s = ζ(s), is considered and phase portraits presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases, the the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest, using the Chebyshev method and using Mathematica. The phase diagrams suggest new analytic properties of zeta, a number of which are proved and a number of which are given in the form of conjectures.

Abstract. The Ramanujan AGM fraction is a construct Rη(a, b) = η + η + a b2 4a 2 η + 9b2 η +... enjoying attractive algebraic properties such as a striking arithmetic-geometric mean (AGM) relation and elegant connections with elliptic-function theory. But the fraction also presents an intriguing computational challenge. Herein we show how to rapidly evaluate R for any triple of positive reals a, b, η, the problematic scenario being when a ≈ b, although even in such cases certain transformations allow rapid evaluation. In this process we find, for example, that when a = b = rational, Rη is essentially an L-series that can be cast therefore as a finite sum of fundamental numbers. We ultimately exhibit an algorithm that yields D good digits of R in O(D) iterations where the implied big-O constant is independent of the positive-real triple a, b, η. Finally, we address the evidently profound theoretical and computational dilemmas that arise when the parameters are allowed to become complex, finding means to extend the AGM relation for complex parameter domains.

We propose means of computing the Fourier expansions of periodic functions appearing in higher moments of the sum-of-digits function and in the solutions of some divide-and-conquer 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.