A mean value function and its computational formula related to D. H. Lehmer's problem

다국어 초록 (Multilingual 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 (Reference)

1 R. K. Guy, "Unsolved Problems in Number Theory" Springer-Verlag 1994

2 L. Carlitz, "The reciprocity theorem for Dedekind sums" 3 : 513-522, 1953

3 W. Zhang, "On the mean values of Dedekind sums" 8 (8): 429-442, 1996

4 C. Jia, "On the mean value of Dedekind sums" 87 (87): 173-188, 2001

5 W. Zhang, "On a problem of D.H.Lehmer and its generalization" 86 (86): 307-316, 1993

6 T. Funakura, "On Kronecker’s limit formula for Dirichlet series with periodic coefficients" 55 (55): 59-73, 1990

7 J. B. Conrey, "Mean values of Dedekind sums" 56 (56): 214-226, 1996

8 L. K. Hua, "Introduction to Number Theory" Science Press 1979

9 T. M. Apostol, "Introduction to Analytic Number Theory" Springer-Verlag 1976

10 W. Zhang, "A problem of D. H. Lehmer and its mean square value formula" 29 (29): 109-116, 2003

11 W. Zhang, "A problem of D. H. Lehmer and its generalization (II)" 91 (91): 47-56, 1994

12 W. Zhang, "A note on the mean square value of the Dedekind sums" 86 (86): 275-289, 2000

13 W. Zhang, "A mean value related to D. H. Lehemer’s problem and the Ramanujan’s sum" 54 (54): 155-162, 2012