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

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Abstract

For squarefree N and x∈R, 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).

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References

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Ferrara, Via Machiavelli, 35, 44100, Ferrara, Italy

    P. Codecà

  2. Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, UK

    M. Nair

Authors
  1. P. Codecà
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  2. M. Nair
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Correspondence to P. Codecà.

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Codecà, P., Nair, M. An extension of a result of Lehmer on numbers coprime to n . Ramanujan J 16, 59–71 (2008). https://doi.org/10.1007/s11139-007-9095-8

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  • DOI: https://doi.org/10.1007/s11139-007-9095-8

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