Abstract
In this paper, we use the mean value theorem of Dirichlet L-functions, the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the two-term exponential sums, and give an interesting identity and asymptotic formula for it.
1 Introduction
Let q be a natural number and h an integer prime to q. The classical Dedekind sums
where
describes the behaviour of the logarithm of the eta-function (see [1, 2]) under modular transformations. The various arithmetical properties of S(h, q) were investigated by many authors, who obtained a series of results, see [3–10]. For example, W. P. Zhang and Y. N. Liu [10] studied the hybrid mean value problem of Dedekind sums and Kloosterman sums
where q ≥ 3 is an integer,
Let p be an odd prime, then one has the identity
where hp denotes the class number of the quadratic field
Let p be an odd prime, then one has the asymptotic formula
where exp(y) = ey.
On the other hand, W. P. Zhang and D. Han [11] studied the sixth power mean of the two-term exponential sums, and proved that for any prime p > 3 with (3, p — 1) = 1, one has the identity
It is natural that one will ask, for the two-term exponential sums
whether there exists an identity (or asymptotic formula) similar to Theorem A(or Theorem B. The answer is yes.
The main purpose of this paper is to show this point. That is, we shall use the mean value theorem of Dirichlet L-functions, the properties of Gauss sums and Dedekind sums to prove the following similar conclusions:
Let p > 3 be an odd prime with (3, p — 1) = 1, then we have the identity
where hp denotes the class number of the quadratic field
Let p > 3 be a prime with (3, p — 1) = 1, then we have the asymptotic formula
It is very interesting that the results in our paper are exactly the same as in reference [10]. This means that there is close relationship between Kloosterman sums and two-term exponential sums. In fact, some close relationships can be found in W. Duke and H. Iwaniec [12].
2 Several lemmas
To complete the proof of our theorems, we need to prove several lemmas. Hereinafter, we shall use some properties of characters mod q and Dirichlet L-functions, all of these can be found in reference [13], so they will not be repeated here.
Let p be an odd prime, a be any integer with (a, p) = 1. For any non-principal character χ mod p, we have the identity
where
From the definition and properties of Gauss sums we have
For any integer a with (a, p) = 1, from Theorem 7.5.4 of [14] we know that
Combining (1) and (2) we have the identity
This proves Lemma 2.1. □
Let p be an odd prime. Then for any non-principal character χ mod p with χ3 ≠ χ0 (the principal character mod p), we have the identity
From Lemma 2.1 the definition and properties of Gauss sums we have
This proves Lemma 2.2 □
Let q > 2 be an integer. Then for any integer a with (a, q) = 1, we have the identity
where L(1,χ) denotes the Dirichlet L-function corresponding to character χ mod d
See Lemma 2 of [9]. □
For any odd prime p, we have the asymptotic formula
See Lemma 6 of [15]. □
3 Proof of the theorems
In this section, we shall complete the proof of our theorems. First we prove Theorem 1.1. From Lemma 2.3. with q = p (an odd prime) we have
and
It is clear that if (3, p — 1) = 1, then for any odd character χ mod p, we have χ3 ≠ χ0, the principal character mod p. Note that
If p ≡ 1 mod 4, then χχ2 ≠ χ0 for all odd character χ mod p. This time, from (4) and (5) we have
If p ≡ 1 mod 4, thenχ2(–1) = — 1. This time we have |τ(χ2χ2)| = 1 and
where we have used the identity
Combining (6) and (7) we may immediately deduce Theorem 1.1.
Now we prove Theorem 1.2. From (3) we have
Now we estimate R1 and R2 in (8) respectively. Note that the identity
from Lemma 2.4 we have the asymptotic formula
If p ≡ 1 mod 4, then there exist two odd characters χ and ŋ such that χŋχ2 =χ0. This time, we have the estimate
So for any prime p > 3, from (4), Lemma 2.2 and Lemma 2.4 we also have the asymptotic formula
Combining (8), (9) and (10) we have the asymptotic formula
This completes the proofs of our results.
Competing interests
The authors declare that they have no competing interests.
Acknowledgement
The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.
This work is supported by N. S. F. (11371291) of P. R. China and G.I.C.F. (YZZ15009) of Northwest University.
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