# A MEAN VALUE FUNCTION AND ITS COMPUTATIONAL FORMULA RELATED TO D. H. LEHMER'S PROBLEM

• Wang, Tingting (College of Science, Northwest A&F University)
• Published : 2016.03.31
• 69 6

#### Abstract

Let p be an odd prime and c be a fixed integer with (c, p) = 1. For each integer a with $1{\leq}a{\leq}p-1$, 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 $1{\leq}a$, $b{{\leq}}p-1$ in which a and $\bar{b}$ are of opposite parity, where $\bar{b}$ is defined by the congruence equation $b{\bar{b}}{\equiv}1$ mod 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.

#### Keywords

Lehmer's problem;error term;mean value;computational formula

#### Acknowledgement

Supported by : N. S. F., P. S. F.

#### References

1. T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
2. L. Carlitz, The reciprocity theorem for Dedekind sums, Pacific J. Math. 3 (1953), 513-522. https://doi.org/10.2140/pjm.1953.3.513
3. J. B. Conrey, E. Fransen, R. Klein, and C. Scott, Mean values of Dedekind sums, J. Number Theory 56 (1996), no. 2, 214-226. https://doi.org/10.1006/jnth.1996.0014
4. T. Funakura, On Kronecker's limit formula for Dirichlet series with periodic coefficients, Acta Arith. 55 (1990), no. 1, 59-73. https://doi.org/10.4064/aa-55-1-59-73
5. R. K. Guy, Unsolved Problems in Number Theory, (Second Edition), Springer-Verlag, New York, 1994.
6. L. K. Hua, Introduction to Number Theory, Science Press, Beijing, 1979.
7. C. Jia, On the mean value of Dedekind sums, J. Number Theory 87 (2001), no. 2, 173-188. https://doi.org/10.1006/jnth.2000.2580
8. W. Zhang, On a problem of D.H.Lehmer and its generalization, Composito Math. 86 (1993), no. 3, 307-316.
9. W. Zhang, A problem of D. H. Lehmer and its generalization (II), Compositio Math. 91 (1994), no. 1, 47-56.
10. W. Zhang, On the mean values of Dedekind sums, J. Theor. Nombres Bordeaux 8 (1996), no. 2, 429-442. https://doi.org/10.5802/jtnb.179
11. W. Zhang, A note on the mean square value of the Dedekind sums, Acta Math. Hungar. 86 (2000), no. 4, 275-289. https://doi.org/10.1023/A:1006724724840
12. W. Zhang, A problem of D. H. Lehmer and its mean square value formula, Japan. J. Math. 29 (2003), no. 1, 109-116. https://doi.org/10.4099/math1924.29.109
13. W. Zhang, A mean value related to D. H. Lehemer's problem and the Ramanujan's sum, Glasg. Math. J. 54 (2012), no. 1, 155-162. https://doi.org/10.1017/S0017089511000498