# Article

Full entry | PDF   (0.2 MB)
Keywords:
Lehmer's problem; error term; Dedekind sums; hybrid mean value; asymptotic formula
Summary:
Let $p$ be an odd prime and $c$ a fixed integer with $(c, p)=1$. For each integer $a$ with $1\le a \leq p-1$, it is clear that there exists one and only one $b$ with $0\leq b \leq p-1$ such that $ab \equiv c$ (mod $p$). Let $N(c, p)$ denote the number of all solutions of the congruence equation $ab \equiv c$ (mod $p$) for $1 \le a$, $b \leq p-1$ in which $a$ and $\overline {b}$ are of opposite parity, where $\overline {b}$ is defined by the congruence equation $b\overline {b}\equiv 1\pmod p$. The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet $L$-functions to study the hybrid mean value problem involving $N(c,p)-\frac {1}{2}\phi (p)$ and the Dedekind sums $S(c,p)$, and to establish a sharp asymptotic formula for it.
References:
[1] Apostol, T. M.: Introduction to Analytic Number Theory. Upgraduate Texts in Mathematics, Springer-Verlag, New York (1976). MR 0434929 | Zbl 0335.10001
[2] Carlitz, L.: The reciprocity theorem for Dedekind sums. Pac. J. Math. 3 (1953), 523-527. DOI 10.2140/pjm.1953.3.523 | MR 0056020 | Zbl 0057.03703
[3] Conrey, J. B., Fransen, E., Klein, R., Scott, C.: Mean values of Dedekind sums. J. Number Theory 56 (1996), 214-226. DOI 10.1006/jnth.1996.0014 | MR 1373548 | Zbl 0851.11028
[4] Guy, R. K.: Unsolved Problems in Number Theory (Second Edition). Unsolved Problems in Intuitive Mathematics 1, Springer-Verlag, New York (1994). MR 1299330
[5] Jia, C. H.: On the mean value of Dedekind sums. J. Number Theory 87 (2001), 173-188. DOI 10.1006/jnth.2000.2580 | MR 1824141 | Zbl 0976.11044
[6] Xu, Z. F., Zhang, W. P.: Dirichlet Characters and Their Applications. Chinese Science Press, Beijing (2008).
[7] Xu, Z. F., Zhang, W. P.: On a problem of D. H. Lehmer over short intervals. J. Math. Anal. Appl. 320 (2006), 756-770. DOI 10.1016/j.jmaa.2005.07.054 | MR 2225991 | Zbl 1098.11050
[8] Zhang, W. P.: A note on the mean square value of the Dedekind sums. Acta Math. Hung. 86 (2000), 275-289. DOI 10.1023/A:1006724724840 | MR 1756252 | Zbl 0963.11049
[9] Zhang, W. P.: A problem of D. H. Lehmer and its mean square value formula. Jap. J. Math. 29 (2003), 109-116. MR 1986866 | Zbl 1127.11338
[10] Zhang, W. P.: On a problem of D. H. Lehmer and its generalization. Compos. Math. 86 (1993), 307-316. MR 1219630 | Zbl 0783.11003
[11] Zhang, W. P.: On the mean values of Dedekind sums. J. Théor. Nombres Bordx. 8 (1996), 429-442. DOI 10.5802/jtnb.179 | MR 1438480 | Zbl 0871.11033

Partner of