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Mathematische Annalen

, Volume 374, Issue 1–2, pp 253–271 | Cite as

Popular subsets for Euler’s \(\varphi \)-function

  • Paul PollackEmail author
Article
  • 65 Downloads

Abstract

Let \(\varphi (n) = \#(\mathbb {Z}/n\mathbb {Z})^{\times }\) (Euler’s totient function). Let \(\epsilon > 0\), and let \(\alpha \in (0,1)\). We prove that for all \(x > x_0(\epsilon ,\alpha )\) and every subset \(\mathscr {S}\) of [1, x] with \(\#\mathscr {S}\le x^{1-\alpha }\), the number of \(n\le x\) with \(\varphi (n)\in \mathscr {S}\) is at most \(x/L(x)^{\alpha -\epsilon }\), where
$$\begin{aligned} L(x) = \exp (\log x\cdot \log _3{x}/\log _2 x). \end{aligned}$$
Under plausible conjectures on the distribution of smooth shifted primes, this upper bound is best possible, in the sense that the number \(\alpha \) appearing in the exponent of L(x) cannot be replaced by anything larger.

Mathematics Subject Classification

Primary 11N64 Secondary 11N25 11N37 

Notes

Acknowledgements

The author is supported by NSF award DMS-1402268. He thanks the referee for a careful reading of the manuscript and helpful comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA

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