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The Ramanujan Journal

, Volume 16, Issue 1, pp 59–71 | Cite as

An extension of a result of Lehmer on numbers coprime to n

  • P. Codecà
  • M. Nair
Article
  • 56 Downloads

Abstract

For squarefree N and xR, define
$$\Delta(x,N)=\sum_{\stackrel{\scriptstyle n\leq xN}{(n,N)=1}}1-x\varphi(N).$$
In the special case when N is composed of primes \(p,p\equiv-1\ (\mathrm{mod}\>q)\) with q>1, Lehmer evaluated \(\Delta(\frac{a}{q},N)\) for any a, 1≤a<q and hence obtained a lower bound for \(\max_{a}|\Delta (\frac{a}{q},N)|\) . We extend this result, for prime q, to N composed of primes p, \(p\equiv r\ (\mathrm{mod}\>q)\) where r is any variable residue modulo q of order congruent to 2 modulo 4. This yields new examples of N for which Δ(N)=sup  x |Δ(x,N)| satisfies Δ(N)≫2ω(N).

Keywords

Euler’s function Discrepancy Lehmer’s bounds 

Mathematics Subject Classification (2000)

11N25 

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References

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    Codecà, P., Nair, M.: Extremal values of \(\Delta(x,N)=\sum_{\stackrel{n\leq xN}{\mbox{\tiny$(n,N)=1$}}}1-x\varphi(N)\) . Can. Math. Bull. 41(3), 335–347 (1998) MATHGoogle Scholar
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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di FerraraFerraraItaly
  2. 2.Department of MathematicsUniversity of GlasgowGlasgowUK

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