
> 
> **Notations** For $f$ a meromorphic function on a domain $\Omega\subseteq \textbf{C}$, we shall say for convenience that $f$ is represented by an Ordinary Dirichlet Series (ODS) if $f$ can be written in the form
>  \begin{equation}
> f(s)=\sum\_{n=1}^{\infty}{\frac{a\_n}{n^s}} \nonumber
> \end{equation}
>  where the series is convergent when $\operatorname{Re}(s)$ is large enough.
> 
> 
> 


I recently found the following result on $L$-functions :


**Theorem**
Let $\gamma$ be a meromorphic function on $\textbf{C}$, $k$ a real. Suppose that $L\_1$ and $L\_2$ are two meromorphic functions on $\textbf{C}$ such that $L\_i$ (for $i=1$ and $2$) satisfies the following conditions


$(i)$ There exists a meromorphic function $L\_i^\*$ on $\textbf{C}$ and a positive real $N\_i$ such that
\begin{equation}
L\_i(k-s)=N\_i^s\gamma(s)L\_i^\*(s) \quad (\forall s\in\textbf{C}), \nonumber
\end{equation}
$(ii)$ there exists a polynomial $P\_i$ such that $P\_iL\_i$ is holomorphic and of finite order in $\textbf{C}$. 


Assume further that $L\_1/L\_2$ has only finitely many poles and that $L\_1/L\_2$ and $L\_1^\*/L\_2^\*$ are represented by ODS.


Then there exist a positive integer $N$ and complex numbers $(a\_d)\_{d|N}$ such that
\begin{equation}
L\_1(s)=\left(\sum\_{u|N}{\frac{a\_u}{u^s}}\right)L\_2(s) \quad (\forall s\in \textbf{C}). \nonumber
\end{equation}


I have also found interesting applications (certainly already known) :


**Corollary $1$** Let $\chi$ and $\varphi$ be two distinct primitive Dirichlet characters. Then $L(\chi,s)/L(\varphi,s)$ has infinitely many poles. 


**Corollary $2$** Let $f$ and $g$ be two linearly independent Hecke forms of integer weight $k$ with respect to $\Gamma\_0(N)$. Then $L(f,s)/L(g,s)$ has infinitely many poles.


I need an answer to those few questions :


$1.$ Is this theorem known ?


$2.$ If yes, does it deserve a publication ?


$3.$ Are there interesting references on that subject ?


I should add that I am a french graduate student (I apologize for my approximative english...) and I worked on number theory on my own : I hope that my problem is not too badly written. Many thanks !



Is this theorem on $L$-functions known?
Closely related problems have been extensively studied; in particular much stronger versions of the corollaries are already known. Here are some references: Fujii was the first to show that a positive proportion of the zeros of two different Dirichlet $L$-functions are 
different. A stronger version of this, namely of finding simple zeros of $L(s,\chi\_1)L(s,\chi\_2)$ was considered by Conrey, Ghosh and Gonek. See this paper of [Bombieri and Hejhal](http://matwbn.icm.edu.pl/ksiazki/aa/aa83/aa8336.pdf) for discussions of these results and some more general results (under some hypotheses). Also of interest is the work of Murty and Murty, *Strong multiplicity one for Selberg's class*, Comptes Rendus 319 (1994). They show that for two different $L$-functions in the Selberg class, the symmetric difference of the multisets of zeros of the two $L$-functions must have $\gg T$ elements up to size $T$. The idea here is that via the explicit formula, the zeros of an $L$-function may be used to identify all its coefficients. 


