(p - 1)th Roots of unity mod pn, generalized Heilbronn sums, Lind-Lehmer constants, and Fermat quotients
Michigan Mathematical Journal(2017)
摘要
For n >= 3, we obtain an improved estimate for the generalized Heilbronn sum Sigma(x=1) (p-1) e(p)(n) (yx(pn-1)) and use it to show that any interval I of points in Zp(n) of length broken vertical bar I vertical bar >> p(1.825) for n = 2, broken vertical bar I vertical bar >> p(2.959) for n = 3, and broken vertical bar I vertical bar >= p(n-3.269)(34/151)(n)+0(1) for n >= 4 contains a (p -1)th root of unity. As a consequence, we derive an improved estimate for the Lind-Lehmer constant for the Abelian group Z(p)(n) and improved estimates for Fermat quotients.
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