A transformation formula related to Dirichlet L -functions with principal character

. We prove a transformation formula for the function for the exponential sum involving the divisor function. This formula can be applied to obtain meromorphic continuation for the Mellin transform of the square of Dirichlet L -function with principal character.


Introduction
Let s = σ + it be a complex variable, χ 0 be the principal character modulo q, q > 1, and let L(s, χ 0 ) = ∞ m=1 χ 0 (m) m s , σ > 1 denote the corresponding Dirichlet L-function. It is well known that the function L(s, χ 0 ) has meromorphic continuation to the whole complex plane with unique simple pole at s = 1 with residue where p denotes a prime number, and ϕ(q) is the Euler totient function. For the investigation of the mean square is needed. Meromorphic continuation of Z 1 (s, χ 0 ) requires a certain transformation formula. Similar transformation formulae are also used in the case of the Riemann zeta-function [6,5]. Let d(m) = d|m 1 denote the divisor function, γ be the Euler constant and Then in [6] and [5], the formulae for Φ(z −1 ) where obtained. Let 1 2 < ̺ < 1, and where Γ (z) denotes the Euler gamma-function, and ζ(s) is the Riemann zeta-function. Then in [4], a formula for Φ ̺ (z −1 ) has been obtained. Let k and l be coprime positive integers, z = 0 and The aim of this note is to obtain a formula for Φ(z −1 , k l ). Let c ′ 0 and c ′′ 0 be the constant terms in [2] for E(s, k l , 0) and E(s, − k l , 0) respectively. Moreover, denote and, for 1 < b < 2, define Theorem 1. Supose that Re z > 0 and Im z = 0. Then
It is not difficult to see that, for σ > max(Re α + 1, 1) This equality show that the function E(s, k l , α) is analytic in the whole complex plane, except for two simple poles at s = 1 and s = 1 + α if α = 0, and a double pole s = 1 if α = 0. Now let k is connected to k by the congruence kk ≡ 1 (mod l). Then equality (1) together with the functional equation for the Lerch zeta-function, see, for example [3], leads to the functional equation for E(s, k l , α) The function E(s, k l , α), for α = 0, was introduced by T. Estermann in [1] for needs of the representation of numbers as a sum of two products. In [2], the extension for α ∈ [−1, 0] was given.
The series of definition of Φ(z, k l ) contains the product d(m) exp{2πi k l } which are coefficients of the Dirichlet series for E(s, k l , 0). Therefore, the function E(s, k l , 0) involved in the formula for Φ(z −1 , k l ).

Proof of the transformation formula
We start with the well-known Mellin formula The later formula together with definition gives the equality 16

A. Balčiūnas
Now we move the line of integration in (3) to the left. Let 0 < a < 1. Since, as it is noted in Section 1, the function E(ω, k l , 0) has a double pole at s = 1, we obtain that Clearly, Therefore, in view of (2), This, (4) and the definition of Φ(z, k l ) show that Hence, Applying the functional equation for the Estermann zeta-function, we find that E 1 − ω, k l , 0 = 1 π 2π l 2−2ω Γ 2 (ω) E ω, k l , 0 + cos(πω)E ω, − k l , 0 .