An extension of a result of Lehmer on numbers coprime to n
Tóm tắt
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)$ 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).
Tài liệu tham khảo
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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)
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