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Suggested Reading
G H Hardy and E M Wright,An Introduction to the Theory of Numbers, Oxford University Press, 1959. See Chapter VIII, and in particular Theorems 332 and 333. The latter theorem attributed to Gegenbauer (1885) says that if Q(x) is the number of square-free numbers not exceedingx, then\(Q(x) = \frac{{6x}}{{\pi ^2 }} + O(\sqrt x ).\) Here O(√x) represents a function whose absolute value is bounded byA√x for some constantA. Use this formula, with a computer program for testing whether a number is square-free, to obtain the value ofπ up to the third decimal place.
P J Davis and R Hersch,The Mathematical Experience, Birkhauser, 1981. We have borrowed our main argument from the discussion on page 366 here. This occurs in a chapter titledThe Riemann Hypothesis where the authors present an argument showing that this most famous open problem in mathematics has an affirmative solution with probability one.
M Kac,Enigmas of Chance, University of California Press, 1987. See Chapters 3, 4, and in particular pages 89–91 of this beautiful autobiography for deeper connections between number theory and probability found by its author. See also his bookStatistical Independence in Probability, Analysis and Number Theory, Mathematical Association of America, 1959.
W Dunham,Journey Through Genius, Penguin Books, 1991. See Chapter 9 titledThe Extraordinary Sums of Leonhard Euler for an entertaining history of the formula (3).
R Bhatia,Fourier Series, Hindustan Book Agency, Second Edition, 2003. See Chapter 3 for several series and products that lead toπ.
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Bhatia, R. The unexpected appearance of Pi in diverse problems. Reson 8, 34–43 (2003). https://doi.org/10.1007/BF02837867
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DOI: https://doi.org/10.1007/BF02837867