Positive integers possessing a weak order

For a positive integerm,we call a positive integertwhich possesses a weak order(modm),if there doesnot exist a positive integer nsuch thattn 1≡ t(modm). Letg(m)be the number of positive integerstforwhich1≤ t≤ mandtpossess a weak order(modm).Forx≥3,letG(x) =∑m≤xg(m).In 1981, V.S. Joshi1]first proved the following as-ymptotic formula:G(x) =αx2 O(x(logx)3), (1)whereα=2ζ(12)∑n∞ =1C(nn2), C(n) =∏p npp 1.In 1986, Yang Zhaohua2]improved (1) toG(x) =αx2 O(x(logx)2). (2) The key ne…

Positive integers possessing a weak order

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) )tn+1 ≡ t(modm).For a number x ≥ 3, letG(x) = ∑m≤xg(m).In this paper, we obtain the asymptotic formula:G(x) = αx2 + O(xlogx),as x→∞.Our result improves the corresponding result with an error term O(xlog2x) of Yang Zhaohua obtained in 1986.