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Hybrid mean value results for a generalization on a problem of D.H. Lehmer and hyper-Kloosterman sums. (English) Zbl 1147.11043

Let \(q>2\) and \(c\) be two integers with \((c,q)=1\),
\[ M(q,c)= \mathop{\sum^q_{\substack{ a=1\\ (a,q)=1}}\;\sum^q_{\substack{ b=1\\ (b,q)=1}}}_{\substack{ ab\equiv c\bmod q\\ 2\nmid a+b}} 1 \]
and \(F(q,c)=M(q,c)-\frac{1}{2}\varphi (q)\).
In [“On a problem of D. H. Lehmer and Kloosterman sums”, Monatsh. Math. 139, No. 3, 247–257 (2003; Zbl 1094.11028)], W. Zhang found that there exist relation between the error term \(F(q,c)\) and the classical Kloosterman sums
\[ K(m,n,q)=\sum_{\substack{ b=1\\ (b,q)=1}}^q e\bigg(\frac{mb+m\bar{b}}{q}\bigg), \] where \(e(y)=e^{2\pi i y}\) and \(\bar{b}\) is defined by equation \(b\bar{b}\equiv 1\bmod q\). Namely, he obtained that
\[ \sum_{\substack{ c=1\\ (c,q)=1}}^q F(q,c)K(\bar{4}c,1,q)=\frac{4}{\pi ^2}\,q\varphi (q)\prod_{p\parallel q}\bigg(1-\frac{1}{p(p-1)}\bigg)+O(q^{3/2+\varepsilon}), \]
where \(\varepsilon\) is any fixed positive number, \(\prod _{p||q}\) denotes the product over all prime divisors \(p\) of \(q\) with \(p|q\) and \(p^2\nmid q\).
In this paper a generalization on this problem is considered. Let
\[ N(q,k,c)= \mathop{\sum^q_{\substack{ a_1=1\\ (a_1,q)=1}} \cdots \sum^q_{\substack{ a_k=1\\ (a_k,q)=1}} \sum^q_{\substack{ b=1\\ (b,q)=1}}}_{\substack{ a_1\dots a_kb\equiv c\bmod q\\ 2\nmid a_1+\cdots+ a_k+b}} (a_1+\cdots+ a_k-b)^2, \]
\(k\geq 1\) and
\[ E(q,k,c)= N(q,k,c)- \frac{3K^2-5k+4}{24} \varphi ^k(q)q^2- \frac{k+1}{12} \varphi^{k-1}(q)q \prod_{p|q}(1-p)\,. \]
Using the Fourier expansion for character sums and the mean value theorems of Dirichlet \(L\)-functions, for any odd \(q\geq 3\) the authors prove that
\[ \sum_{\substack{ c=1\\ (c,q)=1}}^q E(q,k,c)K(\bar{2}^{k+1}c,k,q)= \frac{c_kq^{k+2}\varphi (q)}{\pi ^{k+3}}\prod_{p\parallel q} \bigg(1-\frac{p^k-1}{p^k(p-1)^2}\bigg) +O(q^{k+5/2+\varepsilon}), \]
where \(\varepsilon\) is any positive integer and
\[ c_k=\begin{cases} -6, &\text{if }k=1, \\ i^{k+3}2^{2k-2}[\pi ^2(k^2-k+2)-8(k+1)], &\text{otherwhise.} \end{cases} \]

MSC:

11L05 Gauss and Kloosterman sums; generalizations
11N37 Asymptotic results on arithmetic functions

Citations:

Zbl 1094.11028
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Full Text: Euclid

References:

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