Geodesic
On Lehmer's problem and Dedekind sums
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 909-916
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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.
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.
DOI : 10.1007/s10587-011-0058-2
Classification : 11F20, 11L40, 11M20
Keywords: Lehmer's problem; error term; Dedekind sums; hybrid mean value; asymptotic formula 
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Pan, Xiaowei; Zhang, Wenpeng. On Lehmer's problem and Dedekind sums. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 909-916. doi: 10.1007/s10587-011-0058-2

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