1. Introduction

We consider the argument of Dirichlet L-functions. Let be the Dirichlet L-function, where is a complex variable, associated with a primitive Dirichlet character We denote the non-trivial zeros of by where and are real numbers. Then, when t is not the ordinate of a zero of , we define

This is given by continuous variation along the straight line as varies from to , starting with the value zero. Also, when t is the ordinate of a zero of , we define

In Selberg ^[2], it is known that

and under the generalized Riemann hypothesis (GRH)

Concerning the constant in O-notation, the indication that, for , a formula is valid uniformly in

The purpose of the present article is to prove the following result.

Theorem 1. Assuming GRH. Then, for q > 1

The constant 0:804 obviously does not depend on . And we don't know anything concerning the optimality. Also, the implied constant of the error term does not depend on q. The details of the argument concerning error terms can be seen in the proof of this theorem. However, our result does not include the case of the function S(t) which is defined by the argument of the Riemann zeta-function since we assume q > 1. An explicit upper bound of the function S(t) is obtained by A. Fujii ^[1], where the value is 0.83.

The basic policy of the proof of this theorem is based on A. Fujii ^[1]. In the proof, is seperated by three parts and . Fujii's idea of ^[1] is applied to all parts. But we need Lemma 1, which is an explicit formula for . This lemma is an analogue of Selberg's result.

But the constant part of Theorem 1 is not best result for . In 2007, D. A. Goldston, S. M. Gonek ^[7] proved following result;

for under the Riemann Hypothesis. This result suggest that analogous bound could be true also for the Dirichlet L-function.

Also, E. Carneiro, V. Chandee, M. B. Milinovich ^[5] proved

and

in 2013.

Thus, for the Riemann zeta-function, Fujii's result was improved by D. A. Goldston, S. M. Gonek ^[7], after that the result was improved by E. Carneiro, V. Chandee, M. B. Milinovich ^[5]. Moreover, for a vast class of general L-function (in particular, for ), those better constant are established by E. Carneiro, V. Chandee, M. B. Milinovich ^[6] under GRH.

So, the main result of this paper is a particular case of the work ^[6]. Therefore, Theorem 1 is given by an alternative proof in the case of Dirichlet L-function with a weaker constant using a different method.

In future, we will approach to obtain an explicit upper bound of the multiple integral of the Dirichlet L-function under the GRH based on ideas of ^[5].

2. Some Notations and a Lemma

To prove our result, we introduce some notations and prove the aforementioned Lemma 1.

Let . We suppose that and Let x be a positive number satisfying . Also, we put

and

with

Using these notations, we prove the following lemma.

Lemma 1. Assume the GRH. Let and such that Then for there exist and such that and we have

This is an analogue of Lemma 2 of A. Fujii ^[1].

Lemma 2. Let if and if Then, for and we have

Lemma 2 is similar to Lemma 15 of Selberg ^[2]. We write here only a sketch of the proof of Lemma 2.

If we have

We consider residues which we encounter when we move the path of integration to the left. At the point , the residue is At the zeros , the residues are At the zeros of the residues are Thus, we obtain Lemma 2.

Proof of Lemma 1. Assume the GRH. In Lemma 2, since for

we have

where . Hence Lemma 2 can be rewritten by

(1)

for

In particular, since for we get

(2)

where

Here, since by p. 46 of Selberg ^[2]

we get for

(3)

By (2) and (3) we have

Inserting the above inequality to (1), we obtain Lemma 1.

3. Proof of Theorem 1

The quantity is separated into the following three parts.

say.

First, we estimate . By Lemma 1 we have

(4)

say. Here,

(5)

say.

Next, applying Lemma 1 to , we get

(6)

say.

Next we estimate . By Lemma 16 of Selberg ^[2] we get

say.

Here, we put If , we see easily. If , we have

since for

Here, by (2) and (3) we get

So,

Hence we have

(7)

say.

Finally, we estimate the sums on right-hand sides of (4), (5), (6) and (7). By definition of we have

Similarly,

So, we see

and

For and taking we have

Therefore we obtain the theorem.

Acknowledgments

I thank Prof. Kohji Matsumoto for his advice and patience during the preparation of this paper. I also thank Prof. Giuseppe Molteni, Prof. Yumiko Umegaki and Dr. Ryo Tanaka, who gave many important advice. Finally, I thank the referee who indicates errors in this paper.

References

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[2]	A. Selberg, Contributions to the theory of Dirichlet's L-function, Avh. Norske Vir. Akad. Oslo I:1, (1946), No. 3, 1-62.
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[3]	A. Selberg, Collected Works, vol I, 1989, Springer.
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[4]	E. C. Titchmarsh, The theory of the Riemann zeta-function, Second Edition; Revised by D. R. Heath-Brown. Clarendon Press Oxford, 1986.
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[5]	E. Carneiro, V. Chandee, M. B. Milinovich, Bounding S(t) and S1(t) on the Riemann hypothesis, Math. Ann. Vol. 356, Issue 3 (2013), pp. 939-968.
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