## 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

### Keywords:

Lehmer problem; Kloosterman sums; mean value

Zbl 1094.11028
Full Text:

### References:

  W. Zhang: On the difference between an integer and its inverse modulo $$n$$ , J. Number Theory 52 (1995), 1–6. · Zbl 0826.11002  W. Zhang: On the difference between an integer and its inverse modulo $$n$$ (II), Sci. China Ser. A 46 (2003), 229–238. · Zbl 1217.11084  R.K. Guy Unsolved Problems in Number Theory, Second edition, Springer, New York, 1994. · Zbl 0805.11001  W. Zhang: A problem of D.H. Lehmer and its Generalization (I), Compositio Math. 86 (1993), 307–316. · Zbl 0783.11002  W. Zhang: A problem of D.H. Lehmer and its Generalization (II), Compositio Math. 91 (1994), 47–56. · Zbl 0798.11001  W. Zhang: On the difference between a D.H. Lehmer number and its inverse modulo $$q$$ , Acta Arith. 68 (1994), 255–263. · Zbl 0826.11003  W. Zhang: A problem of D.H. Lehmer and its mean square value formula , Japan. J. Math. (N.S.) 29 (2003), 109–116. · Zbl 1127.11338  W. Zhang, X. Zongben and Y. Yuan: A problem of D.H. Lehmer and its mean square value formula , J. Number Theory 103 (2003), 197–213. · Zbl 1046.11070  W. Zhang: On a problem of D.H. Lehmer and Kloosterman sums , Monatsh. Math. 139 (2003), 247–257. · Zbl 1094.11028  L.J. Mordell: On a special polynomial congruence and exponential sums ; in Calcutta Math. Soc. Golden Jubilee Commemoration Vol., Part 1, Calcutta Math. Soc., Calcutta., 1963, 29\nobreakdash–32. · Zbl 0114.26301  D. Bump, W. Duke, J. Hoffstein and H. Iwaniec: An estimate for the Hecke eigenvalues of Maass forms , Internat. Math. Res. Notices (1992), 75–81. · Zbl 0760.11017  W. Luo, Z. Rudnick and P. Sarnak: On Selberg’s eigenvalue conjecture , Geom. Funct. Anal. 5 (1995), 387–401. · Zbl 0844.11038  R.A. Smith: On $$n$$-dimensional Kloosterman sums , J. Number Theory 11 (1979), 324–343. · Zbl 0409.10024  T.M. Apostol: Introduction to Analytic Number Theory, Springer, New York, 1976. · Zbl 0335.10001  T. Funakura: On Kronecker’s limit formula for Dirichlet series with periodic coefficients , Acta Arith. 55 (1990), 59–73. · Zbl 0654.10039  G. Pólya: Über die Verteilung der quadratishen Reste und Nichtreste , Gottinger Nachrichten (1918), 21–29. · JFM 46.0265.02  C. Pan and C. Pan: Goldbach Conjecture, Science Press, Beijing, 1992. · Zbl 0849.11080  W. Zhang: On a Cochrane sum and its hybrid mean value formula , J. Math. Anal. Appl. 267 (2002), 89–96. · Zbl 1106.11303  W. Zhang, Y. Yi and X. He: On the $$2k$$-th power mean of Dirichlet $$L$$-functions with the weight of general Kloosterman sums , J. Number Theory 84 (2000), 199–213. · Zbl 0958.11061
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