Hybrid mean value on the difference between an integer and its inverse modulo \(q\)
This paper continues previous studies by the second author [J. Number Theory 52, No. 1, 1--6 (1995; Zbl 0826.11002)] on \[ M(n,k)= \sum^n _{\substack{ a=1\\ (a,n)=1}} (a- \overline a)^{2k}, \] \textit{Tianping Zhan} and the second author [Proc. Japan Acad., Ser. A 81, No. 1, 7--11 (2005; Zbl 1088.11072)] on \[ M(q,k,c)=\sum_{\substack{ 1\leq a,b\leq q\\ (a,q)=1,ab\equiv c\pmod q}}(a-b)^{2k}, \] and the second author [Sci. China, Ser. A 46, No. 2, 229--238 (2003; Zbl 1217.11084)] where he used the properties of Dedekind sums and Cochrane sums to give a sharp mean-value formula for \(F(q,1,c)\). For a general integer \(k\geq 2\), let \(F(q,k,c)\) denote the error term in \(M(q,k,c)\), i.e. \[ M(q,k,c)=\frac 1{(2k+1)(k+1)} \varphi(q)q^{2k}+F(q,k,c). \] In [Sci. China (loc. cit.)], the second author asked to find whether there exists an asymptotic formula for the mean value \(\sum_{c=1}^q F^2(q,k,c)\). Interestingly there exists some relation between the error term in \(M(q,k,c)\) and Kloosterman sums \[ K(m,n;q)=\sideset\and {'}\to\sum_{b=1}^q e\left({mb+n\overline b}\over {q}\right), \] where \(e(y)=e^{2\pi iy}\) and \(\overline b\) is defined by the equation \(b\overline b \equiv 1 \bmod q\). Let \[ M(q,k,c)=\frac 1{\varphi(q)}\sideset\and {'}\to\sum_{a=1}^q\sideset\and {'}\to\sum_{b=1}^q (a-b)^{2k}+E(q,k,c). \] It is obvious that \(E(q,1,c)=F(q,1,c)\) and \(E(q,k,c)=F(q,k,c) + O(q^{2k})\), \(k\geq 2\). In this paper, the authors use the generalized Bernoulli numbers, Gauss sums and the mean-value theorems of Dirichlet L-functions to study the hybrid mean value \[ \sideset\and {'}\to\sum_{c=1}^q E(q,k,c)K(c,1;q), \] and give a sharp asymptotic formula. Here \(\sideset\and {'}\to\sum_{c=1}^q\) denotes the summation over all \(a\) such that \((a, q)=1\).
scientific article; zbMATH DE number 5357908