A mean value function and its computational formula related to D. H. Lehmer's problem
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Let $p$ be an odd prime and $c$ be 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 $ab \equiv c \bmod p$. Let $N(c, p)$ denote the number of all solutions of the congruence equation $ab \equiv c \bmod p$ 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 \bmod p $. The main purpose of this paper is using the mean value theorem of Dirichlet $L$-functions and the properties of Gauss sums to study the computational problem of one kind mean value function related to $E(c, p)=N(c, p)-\frac{1}{2}\phi(p)$, and give its an exact computational formula.
Abstract
Let $p$ be an odd prime and $c$ be 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 $ab \equiv c \bmod p$. Let $N(c, p)$ denote the number of all solutions of the congruence equation $ab \equiv c \bmod p$ 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 \bmod p $. The main purpose of this paper is using the mean value theorem of Dirichlet $L$-functions and the properties of Gauss sums to study the computational problem of one kind mean value function related to $E(c, p)=N(c, p)-\frac{1}{2}\phi(p)$, and give its an exact computational formula.
참고문헌 (13)
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