СБОРНИК ДЛЯ 𝐿 -ФУНКЦИЙ THE LAPLACE TRANSFORM OF DIRICHLET 𝐿 -FUNCTIONS

Let 𝜒 be a Dirichlet character modulo 𝑞 . The Dirichlet 𝐿 -function 𝐿 ( 𝑠, 𝜒 ) is defined in the half-plane 𝜎 > 1 by the series and has a meromorphic continuation to the whole complex plane. If 𝜒 is a non-principal character, then the function 𝐿 ( 𝑠, 𝜒 ) is entire one. In the case of the principal character, the function 𝐿 ( 𝑠, 𝜒 ) has unique simple pole at the point 𝑠 = 1 . Dirichlet 𝐿 - functions play an important role in the investigations of the distribution of prime numbers in arithmetical progresions, therefore, their analytic properties deserve a constant attention. In applications, often the moments of Dirichlet 𝐿 -functions are used, whose asymptotic behaviour is very complicated. For investigation of moments, various methods are applied, one of them is based on the application of Mellin transforms. On the other hand, Mellin transforms use Laplace transforms. In the paper, the formulae for the Laplace transform of the function | 𝐿 ( 𝑠, 𝜒 ) | 2 in the critical strip are obtained. They extend the formulae obtained in [3] on the critical line 𝜎 = 12 .


Introduction
Let = + be a complex variable. The Laplace transform L( , ) of a function is defined by provided that the integral exists for > 0 with some 0 ∈ R . It is well known that Laplace transforms are very useful integral transforms having applications in various fields of mathematics and in practice. Analytic number theory is not an exception, here Laplace transforms are applied for the investigation of mean values (moments) of zeta and -functions. The classical example given in the monograph [15] says that if ( ) ≥ 0 for ∈ (0, ∞), and, for some ≥ 0, as −→ ∞. This has been applied for the moments of the Riemann zeta-function ( ) with = 1 and = 2 in [15] and [1]. We remind that the function ( ) is defined, for > 1, by the series and by analytic continuation elsewhere. We observe that more precise formulae for moments (1) require those for L( , | | 2 ). In [9], applications of Laplace transforms for mean values of more general Dirichlet series are given. Let ( ) be the number of divisors of , denote the Euler constant, and let where " ′ "means that the last term in the sum is to be halved if is an integer. In [6], an asymptotic formula for the Laplace transform was obtained. In [5], the Laplace transform was applied to give a simple proof for the classical Voronoi identity where 1 and 1 are the Bessel functions. A very good survey on applications of the Laplace transforms in the theory of the Riemann zeta-function is given in [7]. We remind one more formula used in [1] and [12] for the investigation of the mean square of ( ) on the critical line = 1 2 , namely, where the function ( ) is analytic in the strip | | < . Moreover, in any fixed strip | | with 0 < < , the estimate is true. In [10], the above formula was extended to the critical strip, i.e., the formula for ∫︀ ∞ 0 | ( + )| 2 − with a fixed 1 2 < < 1 has been obtained. Now let be a Dirichlet character modulo , and let ( , ) denote the corresponding Dirichlet -function defined, for > 1, by the series If is a non-principal character, then ( , ) is analytically continuable to an entire function, while if 0 the principal character modulo , then where denotes a prime number, i.e. ( , 0 ) can be analytically continued to the whole complex plane, except for a simple pole at the point = 1 with residue Analytic theory of Dirichlet -functions can be found in [4], [8] and [11]. In [3], the formulae for the Laplace transform were obtained. This note is a continuation of [3], and is devoted to the Lapkace transform where , 1 2 < < 1, is a fixed number. For the statement of the results, we need some notation. Denote by ( ) the Gauss sum, i.e., As usual, denote by Γ( ) the Euler gamma-function, and by ( ) the M¨bius function. Moreover, is the generalized divisor function.
Note that if = 1, then the formula of Theorem 2 implies that of [10].

Lemmas
We remind the following results on the functions ( , ) and ( ). Lemma 1. If is a primitive character modulo , then For the proof, see [13]. Lemma 2. The function ( ) satisfies the functional equation Proof of lemma is given, for example, in [15]. The lemma is Theorem 2.18 in [2]. Now we recall the Mellin formula. Lemma 5. Suppose that > 0. Then Proof of the formula can be found in [14].
Proof. For > max{1, ℜ( + 1)}, we have that The case of the Riemann zeta-function can be found in [15].

Proof of Theorems
It is sufficient to prove the theorems for a slightly different integrals. Доказательство. [Proof of Theorem 1] Consider the function Suppose = 0, then we find that If = 1, similarly as above, we find that By estimates for ( , ) from Lemma 3, both these integrals are uniformly convergent on compact subsets of the strip { ∈ C : | | < }, thus, the function ( , ) is analytic in this region. Suppose that | | , where 0 < < is fixed. If | | is small, then the integrals in (3) and(4) are bounded. If | | is large, then integrating by parts and using the estimate from Lemma 3, we obtain that ( , ) = (| | −1 ).
So we have that, for | | , 0 < < , From (2), we deduce that Therefore, using Lemma 1, we find that Now we move the line of integration in (5) to the right and use Lemma 5. If is a non-principal character, the integrand in (5) is a regular function. Therefore, by Lemma 6, Then Since Hence, in view of (6), we find Using Lemma 4, we write the products in (8) Now we move the line of integration in the last integral to the right. The integrand in (9) has simple poles at the points = 1 and = 2 . Clearly, and Finally, having in mind formulae (10), (11) and Lemma 6, we deduce from (9) that