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Probabilistic Number Theory

In: The Mathematical Legacy of Srinivasa Ramanujan

Author

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  • M. Ram Murty

    (Queen’s University, Department of Mathematics and Statistics)

  • V. Kumar Murty

    (University of Toronto, Department of Mathematics)

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.

Suggested Citation

  • M. Ram Murty & V. Kumar Murty, 2013. "Probabilistic Number Theory," Springer Books, in: The Mathematical Legacy of Srinivasa Ramanujan, edition 127, chapter 0, pages 149-153, Springer.
    • Handle: RePEc:spr:sprchp:978-81-322-0770-2_11
    DOI: 10.1007/978-81-322-0770-2_11
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