On Lehmer's problem and Dedekind sums

@article{Pan2011,
abstract = {Let $p$ be an odd prime and $c$ a fixed integer with $(c, p)=1$. For each integer $a$ with $1\le a \le p-1$, it is clear that there exists one and only one $b$ with $0\le b \le p-1$ 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 $1 \le a$, $b \le p-1$ in which $a$ and $\overline\{b\}$ are of opposite parity, where $\overline\{b\}$ is defined by the congruence equation $b\overline\{b\}\equiv 1\hspace\{4.44443pt\}(\@mod \; p)$. 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.},
author = {Pan, Xiaowei, Zhang, Wenpeng},
journal = {Czechoslovak Mathematical Journal},
keywords = {Lehmer's problem; error term; Dedekind sums; hybrid mean value; asymptotic formula; Lehmer's problem; error term; Dedekind sum; hybrid mean value; asymptotic formula},
language = {eng},
number = {4},
pages = {909-916},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Lehmer's problem and Dedekind sums},
url = {http://eudml.org/doc/196387},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Pan, Xiaowei
AU - Zhang, Wenpeng
TI - On Lehmer's problem and Dedekind sums
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 4
SP - 909
EP - 916
AB - Let $p$ be an odd prime and $c$ a fixed integer with $(c, p)=1$. For each integer $a$ with $1\le a \le p-1$, it is clear that there exists one and only one $b$ with $0\le b \le p-1$ 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 $1 \le a$, $b \le p-1$ in which $a$ and $\overline{b}$ are of opposite parity, where $\overline{b}$ is defined by the congruence equation $b\overline{b}\equiv 1\hspace{4.44443pt}(\@mod \; p)$. 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.
LA - eng
KW - Lehmer's problem; error term; Dedekind sums; hybrid mean value; asymptotic formula; Lehmer's problem; error term; Dedekind sum; hybrid mean value; asymptotic formula
UR - http://eudml.org/doc/196387
ER -