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


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).


Euler’s function Discrepancy Lehmer’s bounds 

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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
  2. 2.
    Codecà, P., Nair, M.: Links between \(\Delta(x,N)=\sum_{\stackrel{n\leq xN}{\mbox{\tiny$(n,N)=1$}}}1-x\varphi(N)\) and character sums. Boll. Unione Mat. Ital. 6(8), 509–516 (2003) MATHGoogle Scholar
  3. 3.
    Codecà, P., Nair, M.: The lesser-known Δ-function in number theory. Am. Math. Mon. 112(2), 131–140 (2005) MATHCrossRefGoogle Scholar
  4. 4.
    Erdös, P.: On a problem in elementary number theory. Math. Stud. XII, 32–33 (1949) Google Scholar
  5. 5.
    Lehmer, D.H.: The distribution of totatives. Can. J. Math. 7, 347–357 (1955) MathSciNetMATHGoogle Scholar
  6. 6.
    Vijayaraghavan, T.: On a problem in elementary number theory. J. Indian Math. Soc. 15, 51–56 (1951) MathSciNetMATHGoogle Scholar
  7. 7.
    Washington, L.C.: Introduction to Cyclotomic Fields. Springer, New York (1982) MATHGoogle Scholar

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

Personalised recommendations