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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 base-2 normality—namely bit randomness in a specific technical(More)
Pursuant to the authors' previous chaotic-dynamical model for random digits of fundamental constants [5], we investigate a complementary, statistical picture in which pseu-dorandom number generators (PRNGs) are central. Some rigorous results are achieved: We establish b-normality for constants of the form i 1/(b m i c n i) for certain sequences (m i), (n i)(More)
From an experimental-mathematical perspective we analyze " Ising-class " integrals. These are structurally related n-dimensional integrals we call C n , D n , E n , where D n is a magnetic susceptibility integral central to the Ising theory of solid-state physics. We first analyze C n := 4 n! ∞ 0 · · · ∞ 0 1 n j=1 (u j + 1/u j) 2 du 1 u 1 · · · du n u n. We(More)
Computations of irregular primes and associated cyclotomic invariants were extended to all primes up to 12 million using multisectioning/convolution methods and a novel approach which originated in the study of Stickelberger codes (Shokrollahi, 1996). The latter idea reduces the problem to that of finding zeros of a polynomial over Fp of degree < (p− 1)/2(More)
Box integrals—expectations | r| s or | r − q| s over the unit n-cube—have over three decades been occasionally given closed forms for isolated n, s. By employing experimental mathematics together with a new, global analytic strategy, we prove that for each of n = 1, 2, 3, 4 dimensions the box integrals are for any integer s hypergeometrically closed ("(More)
We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the(More)
D > 1, alors le nombre #(|y|, N) de 1 dans le développement de |y| parmi les N premiers chiffres satisfait #(|y|, N) > CN 1/D avec un nombre positif C (qui dépend de y), la minorationétant vraie pour tout N suffisamment grand. On en déduit la transcen-dance d'une classe de nombres réels n≥0 1/2 f (n) quand la fonc-tion f , ` a valeursentì eres, crot(More)
We apply experimental-mathematical principles to analyze integrals C n,k := 1 n! ∞ 0 · · · ∞ 0 dx 1 dx 2 · · · dx n (cosh x 1 + · · · + cosh x n) k+1. These are generalizations of a previous integral C n := C n,1 relevant to the Ising theory of solid-state physics [8]. We find representations of the C n,k in terms of Meijer G-functions and nested-Barnes(More)