Positive integers possessing a weak order

摘要

Let m be a positive integer, g(m) be the number of integers t for which 1≤t≤m and there does not exist a positive integer n satisfying (t=t(n))t~ n+1 ≡t(modm).For a number x≥3, letG(x)=∑m≤xg(m).In this paper, we obtain the asymptotic formula:G(x)=αx^2+O(xlogx),as x→∞. Our result improves the corresponding result with an error term O(xlog^2x) of Yang Zhaohua obtained in 1986.