L-functions

Abstract

Let D be a domain in the complex s-plane. Let $u_1(s), u_2(s), \ldots ,$ be a sequence of holomorphic functions on D such that $u_m(s) \ne 1$, $s \in D$, $m \ge 1$, and that $\sum _{m=1}^{\infty }|u_m(s)|$ converges uniformly on D to a bounded function on D. Then

$$\begin{aligned} \lim _{N \rightarrow \infty } \prod _{m=1}^N (1-u_m(s)),\quad s \in D, \end{aligned}$$

converges uniformly to a holomorphic function u(s) on D such that $u(s) \ne 0$, $s \in D$. Furthermore, if $\{ j_1, j_2, \ldots \}$ is any permutation of $\{ 1,2,\ldots \}$, then $\displaystyle \lim _{N \rightarrow \infty }\prod _{m=1}^N (1-u_{j_m}(s))$ converges to the same u(s) in a similar way. The function u(s) is denoted by

$$\begin{aligned} \prod _{m=1}^{\infty } (1-u_m(s)), \quad s \in D. \end{aligned}$$