An Elliott-type theorem for twists of L-functions of elliptic curves

A Joint Elliott Type Theorem for Twists of L-Functions of Elliptic Curves

“…We start with a joint limit theorem which is a generalization of Lemma 1 from [12]. We extend the function ω(p) to the set N by the formula…”

“…is absolutely convergent for σ > 3/2. Moreover, the function L E (s) is analytically continued to an entire function, see, for example, [12]. Now let χ be a Dirichlet character modulo q.…”

“…In the sequel, we assume that q is a prime number. Then it was observed in [12] that the function L E (s, χ), as L E (s), is also entire one. Limit theorems for L E (s, χ) with increasing q were began to study in [6], [7], [8], [9], however, only in the half-plane of absolute convergence σ > 3 2 .…”

“…Limit theorems for L E (s, χ) with increasing q were began to study in [6], [7], [8], [9], however, only in the half-plane of absolute convergence σ > 3 2 . In [12], a limit theorem in the space of analytic functions H(D), D = {s ∈ C : σ > 1}, for the function L E (s, χ) has been obtained. For its statement, we need some notation and definitions.…”