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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Values taken many times by Euler's phi-function


Author: Kent Wooldridge
Journal: Proc. Amer. Math. Soc. 76 (1979), 229-234
MSC: Primary 10A20; Secondary 10H30
MathSciNet review: 537079
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Abstract: Let $ {b_m}$ denote the number of integers n such that $ \phi (n) = m$, where $ \phi $ is Euler's function. Erdős has proved that there is a $ \delta > 0$ such that $ {b_m} > {m^\delta }$ for infinitely many m. In this paper we show that we may take $ \delta $ to be any number less than $ 3 - 2\sqrt 2 $.


References [Enhancements On Off] (What's this?)

  • [1] P. Erdős, On the normal number of prime factors of $ p - 1$ and some related problems concerning Euler's $ \phi $-function, Quart. J. Math. Oxford Ser. 6 (1935), 205-213.
  • [2] H. Halberstam and H.-E. Richert, Sieve methods, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974. London Mathematical Society Monographs, No. 4. MR 0424730 (54 #12689)
  • [3] D. G. Kendall and R. A. Rankin, On the number of Abelian groups of a given order, Quart. J. Math., Oxford Ser. 18 (1947), 197–208. MR 0022569 (9,226c)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0537079-1
PII: S 0002-9939(1979)0537079-1
Article copyright: © Copyright 1979 American Mathematical Society