On the mean value of the two-term Dedekind sums
© Xiaoyu and Zhengang; licensee Springer. 2013
Received: 8 November 2013
Accepted: 21 November 2013
Published: 11 December 2013
The main purpose of this paper is, using the properties of Gauss sums, the estimate for character sums and the analytic method, to study the mean value of the two-term Dedekind sums and give an interesting asymptotic formula for it.
where denotes the multiplicative inverse of , that is, . For convenience, we provide if .
Here, we are concerned whether there exists an asymptotic formula for mean value (1). Regarding this problem, it seems that no one has studied it, at least we have not seen any related result before. The problem is interesting because it can reflect some properties of the value distribution of the polynomial Dedekind sums.
The main purpose of this paper is, using the properties of Gauss sums, the estimate for character sums and the analytic method, to study the mean value properties of the two-term Dedekind sums and give an interesting asymptotic formula for (1). That is, we shall prove the following theorem.
where ϵ denotes any fixed positive number.
Remark In this theorem, we only consider the special polynomial . However, we have not found an effective method to study the general polynomial . This problem should be a further study.
is an open problem, where we provide if .
2 Several lemmas
In this section, we shall give several lemmas, which are necessary in the proof of our theorem. Hereinafter, we shall use many properties of character sums and Gauss sums, all of these can be found in references  and . First we have the following.
where is the Legendre symbol.
where denotes the classical Gauss sums, and .
This proves Lemma 1. □
where denotes the summation over all characters with , ϵ denotes any fixed positive number.
Proof See Lemma 5 of . □
where denotes the Dirichlet L-function corresponding to a character .
Proof See Lemma 2 of . □
where denotes the Legendre symbol.
where we have used the fact that . This proves Lemma 4. □
3 Proof of the theorem
This completes the proof of our theorem.
The authors express their gratitude to the referee for very helpful and detailed comments. This work is supported by the N.S.F. (11371291) and P.N.S.F. (2013JZ001) of P.R. China.
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