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Erschienen in:
2013 | OriginalPaper | Buchkapitel
11. Probabilistic Number Theory
verfasst von : M. Ram Murty, V. Kumar Murty
Erschienen in: The Mathematical Legacy of Srinivasa Ramanujan
Verlag: Springer India
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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
for any ϵ>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.
$$\big|\omega(n) - \log \log n\big| < (\log \log n)^{1/2 + \epsilon} $$