Value-distribution of twisted L -functions of normalized cusp forms

. A limit theorem in the sense of weak convergence of probability measures on the complex plane for twisted with Dirichlet character L -functions of holomorphic normalized Hecke eigen cusp forms with an increasing modulus of the character is proved.


Introduction
Let q ∈ N, and let χ(m) denote a Dirichlet character modulo q. Then the twisted L-function L(s, F, χ) attached to the holomorphic normalized Hecke eigen cusp form F (z) of weight κ for the full modular group is defined, in the half-plane σ > κ+1 2 , by the Dirichlet series is the Fourier series expansion for F (z). The function L(s, F, χ) can be analytically continued to an entire function. Also, in the half-plane σ > κ+1 2 , it can be represented by the Euler product over primes p. The complex numbers α(p) and β(p) satisfy α(p)β(p) = 1, β(p) = α(p), and α(p) + β(p) = c(p).
where χ 0 denotes the principal character mod q. For brevity, let

1,
where in place of dots a condition satisfied by a pair (q, χ(mod q)) is to be written. The aim of this note is a generalization to the space (C, B(C)) of limit theorems with an increasing prime modulus q for |L(s, F, χ)| and arg L(s, F, χ) (see, [3] and [4], respectively). We recall that the function is a characteristic transform of the probability measure P on (C, B(C)) and the measure P is uniquely determined by its characteristic transform w(τ, k).
Let P and P n , n ∈ N, be a probability measures on (C, B(C)). We say that P n converges weakly in sense of C to P if P n converges weakly to P as n → ∞, and, additionally,lim n→∞ P n ({0}) = P ({0}).
For τ ∈ R and k ∈ Z, let and, for primes p and l ∈ N, Thus a τ,k (m) and b τ,k (m) are multiplicative arithmetical functions. Let P C be a probability measure on (C, B(C)) defined by the characteristic transform and let the modulus q of χ be prime. Define .
Then the probability measure P Q,C converges weakly in sense of C to the measure P C as Q → ∞.

Proof of Theorem 1
We give a shortened proof of Theorem 1. At first, we define the characteristic transformation w Q (τ, k) of the probability measure P Q,C , and later we give its asymptotic formula. The definition of P Q,C implies that, for τ ∈ R and k ∈ Z, Note that, in view of the Euler product (1) for L(s, F, χ) and Deligne's estimates L(s, F, χ) = 0 for σ > κ+1 2 . For δ > 0, let R = {s ∈ C: σ κ+1 2 + δ}. Since , from (1) we have that, for s ∈ R, Here the multi-valued functions log(1 − z) and (1 − z) −w , w ∈ C\{0}, in the region |z| < 1 are defined by continuous variation along any path in this region from the values log(1 − z)| z=0 = 0 and (1 − z) −w | z=0 = 1, respectively. Using the above notation, we have that, for |z| < 1, Therefore, (4) shows that, for s ∈ R, whereâ τ,k (m) andb τ,k (m) are multiplicative functions defined, for primes p and l ∈ N, byâ For |τ | c and l ∈ N, with a suitable positive constant c 1 depending on c and k, only. This, estimates (3), and (6)-(7) imply, for |τ | c and l ∈ N, the bounds with a positive constant c 2 depending on c and k. Therefore, by the multiplicativity ofâ τ,k (m) andb τ,k (m), for m ∈ N, where d(m) is the classical divisor function. Now we give an asymptotic formula for the characteristic transform w Q (τ, k) as Q → ∞. Let r = log Q. It is well known that d(m) = O ε (m ε ) with every positive ε. Therefore, for s ∈ R , |τ | c and any fixed k ∈ Z, estimates (8) and (9) yield Substituting this in (5), we find that Thus, in view of (2), for s ∈ R, |τ | c and any fixed k ∈ Z, hold. However, (6)-(7) and the definitions of a τ,k (m) and b τ,k (m), show that . Therefore, by (10), for s ∈ R, |τ | c and any fixed k ∈ Z, (11) It is easily seen that, for m = n, m r, as Q → ∞, If (m, q) = 1, then χ=χ(mod q) χ(m)χ(n) = q − 1 if m ≡ n(mod q), 0 if m ≡ n(mod q).