Limit Theorems for Twists of L -Functions of Elliptic Curves. II

. In the paper, a limit theorem for the argument of twisted with Dirichlet character L -functions of elliptic curves with an increasing modulus of the character is proved.


Introduction
In [3], we began to study limit theorems for twisted with Dirichlet character L-functions of elliptic curves with an increasing modulus of the character, and obtained a limit theorem of such a type for the modulus of these twists. Let E be an elliptic curve over the field of rational numbers given by the Weierstrass equation with non-zero discriminant ∆ = −16(4a 3 + 27b 2 ). For each prime p, denote by E p the reduction of the curve E modulo p which is a curve over the finite field F p , and define λ(p) by where |E(F p )| is the number of points of E p . The L-function L E (s), s = σ + it, of the elliptic curve E is defined by the Euler product L E (s) = for all primes, the product defining L E (s) converges uniformly on compact subset of the half-plane {s ∈ C: σ > 3 2 } and define there an analytic function without zeros. Moreover, in [1], the Taniyama-Shimura conjecture has been proved, therefore, the function L E (s) is analytically continued to an entire function, and satisfies the functional equation where, as usual, Γ (s) denotes the Euler gamma-function, N is the conductor of the curve E, and w = ±1.
The twist L E (s, χ) with Dirichlet character χ for the function L E (s) is defined similarly. For σ > 3 2 , we have that and function L E (s, χ) is also analytically continued to an entire function. Suppose that the modulus q of the character χ is a prime number, and is not fixed. Denoting by χ 0 the principal character modulo q, for Q 2, define

1,
where in place of dots we will write a condition satisfied by a pair (q, χ(mod q)). Let B(S) stand for the class of Borel sets of the space S. Then in [3], the weak convergence of the frequency, as Q → ∞, has been obtained. To state a limit theorem, we need some additional notation and definitions. For p ∆, let α(p) and β(p) be conjugate complex numbers such that α(p)β(p) = p and α(p) + β(p) = λ(p). Then (1.2), for σ > 3 2 , can be rewritten in the form (1.3) As in [3], we use the notation η = η(τ ) = iτ /2, τ ∈ R, and, for primes p and k ∈ N, For p ∆ and k ∈ N, we set where α(p) and β(p) denote the conjugates of α(p) and β(p), respectively. For p | ∆ and k ∈ N, we define where p l m means that p l | m but p l+1 m. On (R, B(R)) define the probability measureP by the characteristic transforms [5], Theorem 1 [see [3]]. Suppose that σ > 3 2 . ThenP Q converges weakly toP as Q → ∞.
The other results for L-functions with increasing modulus of the character are shortly discussed in [3].
The aim of this paper is to prove a limit theorem for the argument of the function L E (s, χ). The estimate (1.1) and (1.3) show that L E (s, χ) = 0 for σ > 3 2 . Thus, for σ > 3 2 , arg L E (s, χ) is well defined. For k ∈ Z, let θ = θ(k) = k 2 , for primes p and l ∈ N, and d k (1) = 1. Now similarly to (1.4) and (1.5), for p ∆ and l ∈ N, we define If p | ∆, then, for l ∈ N, we set Moreover, for m ∈ N, we set Thus, a k (m) and b k (m) are multiplicative functions. Denote by γ the unit circle on the complex plane. Furthermore, let P be a probability measure on (γ, B(γ)) defined by the Fourier transform The main result of this paper is the following statement. Then converges weakly to P as Q → ∞.
We recall that a distribution function F (x) is said to be a distribution function mod 1 if Let F n (x), n ∈ N, and F (x) be distribution functions mod 1. We say that F n (x), as n → ∞, converges weakly mod 1 to F (x), if at all continuity points Denote by L(s, χ) the Dirichlet L-functions. Elliott in [2], for σ > 1 2 , obtained the weak convergence mod 1, as Q → ∞, for µ Q 1 2π arg L(s, χ) x(mod 1) .
From Theorem 2, the following corollary follows.
converges weakly mod 1 to the distribution function mod 1 defined by the Fourier transform g(k) as Q → ∞.
Differently from Dirichlet L-functions, we do not have any information on the convergence of the series defining the function L E (s, χ), χ = χ 0 , in the region σ > 1. Therefore, we can prove Theorem 2 only in the half-plane of absolute convergence of the mentioned series. Of course, we have a conjecture that the statement of Theorem 2 remains also true for σ > 1, however, at the moment we can not prove this.

Fourier Transform
Let g Q (k), k ∈ Z, denote the Fourier transform of P Q i.e., g Q (k) = γ x k dP . Then the definition of P Q implies the equality For the proof of Theorem 2, we need the asymptotics of g Q (k) as Q → ∞. In this section, we give an expression for g Q (k) convenient for the investigation of its asymptotics.
For any fixed δ > 0, denote by R the region {s ∈ C: σ 3 2 + δ}. For s ∈ R, we have that Therefore, for s ∈ R and k ∈ Z \ {0}, formula (1.3) yields Here the multi-valued functions log(1 − z) and (1 − z) ±θ in the region |z| < 1 are defined by continuous variation along any path lying in this region from the values log(1 − z)| z=0 = 0 and (1 − z) ±θ | z=0 = 1, respectively. In the disc |z| < 1, by the definition of d k (p l ) we have that Therefore, (2.2) implies that, for s ∈ R, Letâ k (m) andb k (m) be multiplicative functions with respect to m defined, for primes p ∆ and l ∈ N, bŷ and, for primes p | ∆ and l ∈ N, bŷ For l ∈ N, we have that where the constant c depends on k, only. By the definition of α(p) and β(p), we have that |α(p)| = |β(p)| = √ p. Therefore, for p ∆ and l ∈ N, (2.4) and It is known [4] that, for p | ∆, the numbers λ(p) are equal to 1 or 0. Thus, by (2.6)-(2.7) we have that, for p | ∆, |â k (p l )| (l + 1) c , |b k (p l )| (l + 1) c . Therefore, in view of (2.3), we conclude that, for every fixed k ∈ Z \ {0} and and s ∈ R, This and (2.1) give an expresion for the Fourier transform (2.14) 3 Proof of Theorem 2 Having (2.14), we are in position to obtain the asymptotics for g Q (k) as Q → ∞.
First we modify the right-hand side of (2.14). Let c 1 = 2c + 1. Then, using (2.11)-(2.13), we find that, for s ∈ R, any fixed k ∈ Z \ {0} and N ∈ N, Therefore, taking N = log Q, and using the estimate [3] M Q = Q 2 2 log Q + O Q 2 log 2 Q as well as (2.11) and (2.12) type estimates for a k (m) and b k (m), we find that, for any fixed k ∈ Z \ {0} and s ∈ R,