Compositio Mathematica 23(2), 233-250.
ISSN/ISBN: Not available at this time. DOI: Not available at this time.
Abstract: INTRODUCTION: Let E_{i}(n) be the number of even digits and O_{i}(n) the number of odd digits occurring in the ith column of the table of logarithms of the first n integers to the base 10. In 1899 Henri Poincaré expressed the belief that lim_{n}E_{i}(n)/n and lim_{n}O_{i}(n) exist and equal 1/2. Franel disproved Poincaré's conjecture by showing that these limits do not exist. Franel also showed that the sequence of arithmetic means of digits in the ith column has no limit, and gave information on the set of accumulation points of these sequences. We replace Franel's analytic approach by elementary counting arguments. We find the derivation simpler, the ideas transparent, and the methods more general. We generalize to arbitrary base of logarithms, derive the exact set of accumulation points for these sequences, and show that the limit does not exist for many sequences not treated by Franel
Bibtex:
@article {,
AUTHOR = {Thorp, Edward and Whitley, Robert},
TITLE = {Poincare's conjecture and the distribution of digits in logarithm tables},
JOURNAL = {Compositio Mathematica},
YEAR = {1971},
VOLUME = {23},
NUMBER = {2},
PAGES = {233-250},
}
Reference Type: Journal Article
Subject Area(s): Analysis