Probabilistic Number Theory

M. Ram Murty and V. Kumar Murty
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M. Ram Murty: Queen’s University, Department of Mathematics and Statistics
V. Kumar Murty: University of Toronto, Department of Mathematics

Chapter Chapter 11 in The Mathematical Legacy of Srinivasa Ramanujan, 2013, pp 149-153 from Springer

Abstract: Abstract The field of probabilistic number theory has its origins in a famous 1917 paper of Hardy and Ramanujan. In that paper, they studied the “normal order” of the arithmetic function ω(n), defined as the number of distinct prime divisors of n. They showed that with probability one $$\big|\omega(n) - \log \log n\big| 0. In other words, ω(n) is “usually” loglogn. This theorem was later amplified and expanded to cover a galaxy of arithmetical functions by Erdös, Kac, Kubilius, and Elliott, to name a few. In this chapter, we survey this development of probabilistic number theory as well as its link to the theory of modular forms.

Keywords: Probabilistic Number Theory; Kubilius; Arithmetic Functions; Modular Forms; Loglogn (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-81-322-0770-2_11

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