UNIQUENESS OF L - FUNCTIONS IN THE EXTENDED SELBERG CLASS

Abstract

Ritt's Second Theorem described polynomial solutions of the functional equation P (f ) = Q(g), where P, Q are polynomials. In this paper, using techniques of value distribution theory into account the special properties of L - functions, we describe solutions of the above equation  for  L - functions and a class of polynomials of Fermat-Waring type. Namely, use Lemma 2.1, Lemma 2.2, and Lemma 2.5, we study conditions to equations in the Theorem 1.1 have solutions on sets of L - functions in the extended Selberg class. Then we apply the obtained results from the Theorem 1.1, and use Lemma 2.3, Lemma 2.4, and Lemma 2.6 to study the uniqueness problem for L - functions sharing finite set in the Theorem 1.2.