The Ramanujan Journal

, Volume 16, Issue 1, pp 59–71

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

Article

## 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

11N25

## References

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)
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)
3. 3.
Codecà, P., Nair, M.: The lesser-known Δ-function in number theory. Am. Math. Mon. 112(2), 131–140 (2005)
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)
6. 6.
Vijayaraghavan, T.: On a problem in elementary number theory. J. Indian Math. Soc. 15, 51–56 (1951)
7. 7.
Washington, L.C.: Introduction to Cyclotomic Fields. Springer, New York (1982)