Values taken many times by Euler’s phi-function

Wooldridge, Kent

doi:10.1090/S0002-9939-1979-0537079-1

Values taken many times by Euler’s phi-function
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Proc. Amer. Math. Soc. 76 (1979), 229-234 Request permission

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

On the normal number of prime factors of

and some related problems concerning Euler’s

function

6

H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR 0424730
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 22569, DOI 10.1093/qmath/os-18.1.197