1. Introduction
Let
be an integer,
denotes a Dirichlet character
. Then Dirichlet
L-functions
(see [
1]) defined for
by the series
where
s denotes a complex number, and
.
If is any non-principal character , then is an entire function of s. If is the principal character , then is analytic everywhere except for a simple pole at with residue , denotes the Euler function.
As we all know, Dirichlet
L-functions play a very important role in the research of analytic number theory, many famous problems of number theory are closely related to it. For example, the famous Goldbach’s conjecture, the distribution of twins prime and so on. Because of the importance of these functions, many scholars have studied its various properties, and obtained a series of important results. For example, M. Bordignon [
2,
3] studied the explicit bounds on exceptional zeroes of Dirichlet L-functions, and obtained a sharp upper bound estimate.
J. Andrade and S. Baluyot [
4] proved some new results for the small zeros of Dirichlet
L-functions of quadratic characters of prime modulus.
W. P. Zhang [
5] proved that for any integer
, one has the identity
where
denotes the summation over all odd characters
(i.e.,
),
denotes the product over all different prime divisors of
q.
W. P. Zhang [
6] introduced a generalized Dedekind sums, then he used the properties of this generalized Dedekind sums to prove the following identities
and
where
denotes the summation over all even characters
.
X. Lin [
7] also proved a general mean square value formula for Dirichlet
L-functions
. That is, for any positive integer
k, she obtained an exact calculating formula for
But the formulae in [
7] are very long, and they do not look too beautiful. Especially, the calculation of coefficients in the formulae are very complicated. Therefore, in order to save the space, there is no need to list them here.
Some other papers related to
L-functions can be found in references [
8,
9,
10,
11,
12,
13,
14,
15,
16,
16], we do not want to list them all here.
Very recently, W. P. Zhang and D. Han [
17] studied the computational problems of the reciprocal sums of one kind of Chebyshev polynomials, and proved some interesting identities. Some of them are as follows:
Let
q be an odd number with
. Then for any positive integer
k and integer
h with
, one has the identities
and
where
denotes the Riemann zeta-function, and the constants
are defined as
, and
.
The special cases in [
17] has also been studied by Y. K. Ma and X. X. Lv [
18]. That is, they studied the cases
and 3, and obtained some identities.
This paper, as a note of [
7,
17], we will use the elementary methods and the properties of Dirichlet
L-functions to study the computational problem of one kind of special mean square value of Dirichlet
L-functions, and give a new and exact calculating formula for it. As some applications, we obtain two interesting inequalities involving character sums and trigonometric sums. That is, we will prove the following two conclusions:
Theorem 1. Let be an integer and χ denote a Dirichlet character mod q. Then for any integer , we have the identitywhere denotes the principal character mod q, and the constants are defined by , , , ,
, .
Theorem 2. Let q be an integer with . Then for any integer , we have the identity The main difference between our results and X. Lin [
7] lies in the form of the mean square value of
L-functions. Feature of our results is that they are simple in form and easy to calculate. Note that
is the same as
(
) in [
17]. It can be calculated by the recursive formula
for all integers
,
and
. To better understand the consequences of these theorems, we can use Mathematica software to calculate the value of
for all integers
. Here we give partial values of
as shown in the following
Table 1:
If
is the principal character modulo
q with
, then note the identities
and
where
denotes the Bernoulli numbers, and
denotes the product over all distinct prime divisors of
q. Therefore, for any fixed positive integer
k, we can give the exact values in Theorems 1 and 2. Especially for integers
and 2, from these theorems we may immediately deduce the following results:
Corollary 1 ([
6,
7])
. Let q be an integer with , then we have the identity Corollary 2. Let q be an integer with , then we have the identity Corollary 3 ([
6])
. Let q be an integer with , then we have the identity Of course, Corollary 3 is not a corollary of Theorem 1 and 2, but rather a corollary of their proofs (and the lemmas building up to the proofs).
Corollary 4 ([
6,
7])
. Let q be an integer with , then we have the identity Corollary 5. Let q be an integer with , then for any integer h with and χ mod q, we have the estimate Corollary 6. Let q be an integer with , then for any integer h with and χ mod q, we have the estimateEspecially, if p is an odd prime, then we have the estimate Corollary 7. Let denotes the Legendre’s symbol mod 5. Then we have Corollary 8. Let denotes the Legendre’s symbol mod 3. Then we have 3. Proofs of the Theorems
In this section, we shall complete the proofs of our theorems. First if
, then from the first formula in Lemma 3 we have
and
where
denotes the principal character
.
Note that if
, then we have the identity
So from (
15), (
16) and the orthogonality of the characters we have
This proves Theorem 1.
Now we prove Theorem 2. If
be any odd Dirichlet character
, then note the identity
where we have used the fact that the inner sum is 0, if
.
From this identity, (
15), the orthogonality of characters
and the second formula of Lemma 3 we have
This completes the proof of Theorem 2.
Now we prove Corollary 3. From (
1) and Lemma 1 we have
So from (
17) and the orthogonality of the characters
we have
It is clear that (
18) implies the identity
Corollarys 5 and 6 can be deduced from Lemmas 1 and 2. In fact if
, then from Lemma 1 with
we have
or
If
, then we have
Now Corollary 5 follows from (
19) and (
20).
Similarly, we can also deduce Corollary 6.
Corollarys 7 and 8 are the special cases of Theorems 1 and 2. In fact, for any odd prime
p, it is clear that
is a real character
, so
for all positive integers
n. Of course, from [
1] (Theorem 6.20) we can also get
. If we taking
, then note that
and
are all even characters
, so from Theorem 1 we can easily deduce Corollary 7.
Corollary 8 follows from Theorem 2 with .
This completes the proofs of our all results.