## On the Random Character of Fundamental Constant Expansions (2001)

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Venue: | EXPERIMENTAL MATHEMATICS |

Citations: | 51 - 14 self |

### BibTeX

@MISC{Bailey01onthe,

author = {David H. Bailey and Richard E. Crandall},

title = { On the Random Character of Fundamental Constant Expansions },

year = {2001}

}

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### Abstract

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.

### Citations

2192 | The art of computer programming - Knuth - 1998 |

311 | Uniform Distribution of Sequences - Kupiers, Niederreiter - 1974 |

101 | On the rapid computation of various polylogarithmic constants
- Bailey, Borwein, et al.
- 1997
(Show Context)
Citation Context ...o base 2). The algorithmic motivation for our current treatment is the recent discovery of a simple algorithm by which one can rapidly calculate individual digits of certain polylogarithmic constants =-=[2]-=-. This “BBP” algorithm (named from the authors of [2]) has already given rise to a small computational industry of sorts. For example, the quadrillionth binary digit of π, the billionth binary digit o... |

65 | Analysis of PSLQ, an integer relation finding algorithm
- Ferguson, Bailey, et al.
- 1999
(Show Context)
Citation Context ...) = n=0 55n ( ) 5 1 + 5n +2 5n +3 = 25 2 log ⎛ ⎜ ⎝ 781 ( √ 57 − 5 5 256 57 + 5 √ ) 5 √ 5 ⎞ ⎟ ⎠ . This result dampens the hope that a purely experimental mathematics approach (e.g. PSLQ-based numerics =-=[13]-=-) will resolve any polylogarithm form; indeed, to discover the above example one would need to have in one’s basis of possible terms not only quadratic surds as coefficients but also logarithms of suc... |

53 |
Construction of decimals normal in the scale of ten
- Champernowne
- 1933
(Show Context)
Citation Context ...nt, namely the number: C10 =0.123456789101112131415 ..., where the positive integers are trivially concatenated, is known to be normal to base 10, although existing proofs of even this are nontrivial =-=[10, 18]-=-. One can, of course, construct a binary or ternary equivalent of this constant, by concatenating digits in such bases. In a separate treatise we touch upon the theory of continued fractions, noting f... |

45 | Computational strategies for the Riemann zeta function
- Borwein, Bradley, et al.
- 2011
(Show Context)
Citation Context ...l computational industry of sorts. For example, the quadrillionth binary digit of π, the billionth binary digit of log 2 and the hundred-millionth binary digit of ζ(3) have been found in this fashion =-=[2, 5, 7, 19]-=-. Our intent here is not to present new computational results, but instead to pursue the theoretical implications of this algorithm. Let us describe the BBP algorithm by way of example. We start with ... |

41 | Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, Eur
- Broadhurst
- 1999
(Show Context)
Citation Context ... −2, −1, −1, 1, 0)) 2 and π 2 = 9 P (2, 16, 6, (16, −24, −8, −6, 1, 0)), 8 and thus one may address π2 itself within the theory. We should add that a base 3 series is known for π2 , due to Broadhurst =-=[6]-=-: π 2 = 2 P (2, 729, 12, (243, −405, −81, −27, −72, −9, −9, −5, 1)) 27 Similar high-order generalizations can be developed for log 2 2 and for the Catalan constant G as in: G − π 8 1 log 2 = B(2, 2, ,... |

25 | Note on normal numbers - Copeland, Erdös - 1946 |

23 | Topics in Advanced Scientific Computation - Crandall - 1996 |

21 | Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5 - Broadhurst - 1998 |

16 | On a problem of Steinhaus about normal numbers - Cassels - 1959 |

5 | Topics in advanced scienti c computation - Crandall - 1996 |

4 | Analysis of PSLQ, an integer relation nding algorithm - Ferguson, Bailey, et al. - 1999 |

3 | Borwein and Simon Ploue, \On The Rapid Computation of Various Polylogarithmic Constants - Bailey, Peter - 1997 |

3 |
On the Normality of Fundamental Constants
- Lagarias
- 2000
(Show Context)
Citation Context ...theory, chaotic dynamics, ergodic theory, pseudorandom number generation, probability and statistics. Some of these connections are explored in the companion paper [3] and in a manuscript by Lagarias =-=[16]-=-). 2. Nomenclature and fundamentals We denote by ⌊α⌋ and {α} respectively the usual floor and fractional-part extractions of a real α. In general we have α = ⌊α⌋ + {α}, noting that the fractional part... |

3 | A Simple Formula for - Adamchik, Wagon - 1997 |

3 |
On a Problem of Steinhaus about
- Cassels
- 1959
(Show Context)
Citation Context ...se 2, or for that matter to any power-oftwo base. The wording of this latter part is critical: there exist numbers normal to some base b but not to some other base a that is not a rational power of b =-=[9, 15]-=-. For example, the standard Cantor set has members that are normal to base 2, yet none of its members is normal to base 3. Moreover, there are results on the class of “absolutely abnormal” numbers, me... |

2 |
A Simple Formula for Pi
- Adamchik, Wagon
- 1997
(Show Context)
Citation Context ... converting R → R ′ . 15This simple, base-16 prescription for π is not unique—one also has the following, which was first discovered by Ferguson and Hales [2] and independently by Adamchik and Wagon =-=[1]-=-: π = 4B(1, 2, 1 , (1, 1, 1, 0, −1, −1, −1, 0) 2 This formula may be written in the P -form notation as π = 1 P (1, 16, 8, (8, 8, 4, 0, −2, −2, −1, 0)). 4 Actually, various expansions for π arise from... |

2 |
Item 120 in
- Beeler
- 1972
(Show Context)
Citation Context ...ld emerge. As just one example of an interesting departure from Hypothesis A, said departure involving a slowly decaying perturbation function, consider the following expansion for the Euler constant =-=[4]-=-: γ − 1 2 = ∞∑ 1 2k+1 ⎛ k−1 ∑ ( k−j )−1 ⎝−1+ 2 + j j ⎞ ⎠ . k=1 j=0 Here the relevant perturbation function is rk =(1/2)(−1+ ∑ )k and exhibits a slow decay (evidently: rn ∼ 1/ √ n). Needless to say, an... |

2 |
Conjecture on Integer-Base Polylogarithmic Zeros Motivated by the Cunningham Project
- Broadhurst
- 2000
(Show Context)
Citation Context ...nd class of exceptions we call the “Ferguson anomalies,” involving a fascinating and evidently rare phenomenon. These anomalies are also known as “Zagier zeros,” which involve polylogarithmic ladders =-=[8]-=-. We only know of a few genuinely different examples (note that mere translation of indices can turn one example, say a zero sum, into a rational sum giving nothing new). Here are three, where we writ... |

2 |
mathematics web site
- Weisstein
(Show Context)
Citation Context ... a separate treatise we touch upon the theory of continued fractions, noting for the moment that the Champernowne constant is known to have some gargantuan elements in 6its simple continued fraction =-=[20]-=-. Another example of an intentional construction is the Copeland-Erdős number: 0.23571113171923 ..., in which the primes are simply concatenated; this number being likewise normal to base 10 [11]; thi... |

2 |
available from http://xxx.lanl.gov/format/math/9803067
- Broadhurst
- 1998
(Show Context)
Citation Context ...l computational industry of sorts. For example, the quadrillionth binary digit of π, the billionth binary digit of log 2 and the hundred-millionth binary digit of ζ(3) have been found in this fashion =-=[2, 5, 7, 19]-=-. Our intent here is not to present new computational results, but instead to pursue the theoretical implications of this algorithm. Let us describe the BBP algorithm by way of example. We start with ... |

1 | Absolutely Abnormal Numbers," manuscript, 2000, available from http://arXiv.org/ps/math/0006089 - Martin |

1 | The Quadrillionth Bit of Pi is `0'," manuscript, 2000, available from http://cecm.sfu.ca/pihex/announce1q.html - Percival |

1 | Random generators and normal numbers", preprint - Bailey, Crandall - 2001 |

1 | Computational strategies for 190 - Borwein, Bradley, et al. |

1 | Absolutely abnormal numbers ", preprint - Martin - 2000 |

1 | The quadrillionth bit of pi is `0"', preprint, 2000. See http://cecm.sfu.ca/ pihex/announce1q.html - Percival |

1 | Computational strategies for - Borwein, Bradley, et al. - 2000 |

1 | Absolutely abnormal numbers", preprint - Martin - 2000 |

1 |
Absolutely Abnormal Numbers,” manuscript, 2000, available from http://arXiv.org/ps/math/0006089
- Martin
(Show Context)
Citation Context ...meaning numbers not normal to any base. Any rational number is of this class, of course, yet the class is uncountable, and there exist proven, constructive examples of absolutely abnormal irrationals =-=[17]-=-. It is a celebrated theorem of Weyl that α is irrational if and only if the sequence ({nα} : n =1, 2, 3,...) is equidistributed [15, pg. 8]. Note however that in the present treatment we are not conc... |

1 |
The Quadrillionth Bit of Pi is ‘0’,” manuscript, 2000, available from http://cecm.sfu.ca/pihex/announce1q.html
- Percival
(Show Context)
Citation Context ...l computational industry of sorts. For example, the quadrillionth binary digit of π, the billionth binary digit of log 2 and the hundred-millionth binary digit of ζ(3) have been found in this fashion =-=[2, 5, 7, 19]-=-. Our intent here is not to present new computational results, but instead to pursue the theoretical implications of this algorithm. Let us describe the BBP algorithm by way of example. We start with ... |