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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.
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참고문헌 (13)
참고문헌 신청
T. M. Apostol / 1976 / Introduction to Analytic Number Theory / Springer-Verlag
L. Carlitz / 1953 / The reciprocity theorem for Dedekind sums / Pacific J. Math 3:513~522
J. B. Conrey / 1996 / Mean values of Dedekind sums / J. Number Theory 56(2):214~226
T. Funakura / 1990 / On Kronecker’s limit formula for Dirichlet series with periodic coefficients / Acta Arith 55(1):59~73
R. K. Guy / 1994 / Unsolved Problems in Number Theory / Springer-Verlag
L. Carlitz / 1953 / The reciprocity theorem for Dedekind sums / Pacific J. Math 3:513~522
J. B. Conrey / 1996 / Mean values of Dedekind sums / J. Number Theory 56(2):214~226
T. Funakura / 1990 / On Kronecker’s limit formula for Dirichlet series with periodic coefficients / Acta Arith 55(1):59~73
R. K. Guy / 1994 / Unsolved Problems in Number Theory / Springer-Verlag
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