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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
P.D.T.A. Elliott, Probabilistic Number Theory, I, II (Springer, New York, 1980)
P. Erdös, M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions. Am. J. Math. 62(1), 738–742 (1940)
P. Erdös, C. Pomerance, On the normal number of prime factors of ϕ(n). Rocky Mt. J. Math. 15, 343–352 (1985)
G.H. Hardy, S. Ramanujan, The normal number of prime factors of a number n. Q. J. Math. 48, 76–92 (1917)
Y.R. Liu, M.R. Murty, The Turán sieve method and some of its applications. J. Ramanujan Math. Soc. 14(1), 21–35 (1999)
Y.R. Liu, M.R. Murty, Sieve methods in combinatorics. J. Comb. Theory, Ser. A 111(1), 1–23 (2005)
Y.R. Liu, M.R. Murty, A weighted Turán sieve method. J. Number Theory 116(1), 1–20 (2006)
M.R. Murty, V.K. Murty, Prime divisors of Fourier coefficients of modular forms. Duke Math. J. 51, 57–76 (1984)
M.R. Murty, V.K. Murty, An analogue of the Erdös–Kac theorem for Fourier coefficients of modular forms. Indian J. Pure Appl. Math. 15, 1090–1101 (1984)
M.R. Murty, F. Saidak, Non-abelian generalizations of the Erdös–Kac theorem. Can. J. Math. 56(2), 356–372 (2004)
P. Turan, On a theorem of Hardy and Ramanujan. J. Lond. Math. Soc. 9, 274–276 (1934)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer India
About this chapter
Cite this chapter
Murty, M.R., Murty, V.K. (2013). Probabilistic Number Theory. In: The Mathematical Legacy of Srinivasa Ramanujan. Springer, India. https://doi.org/10.1007/978-81-322-0770-2_11
Download citation
DOI: https://doi.org/10.1007/978-81-322-0770-2_11
Published:
Publisher Name: Springer, India
Print ISBN: 978-81-322-0769-6
Online ISBN: 978-81-322-0770-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)