# Is this theorem on $L$-functions known?

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 $$f(s)=\sum_{n=1}^{\infty}{\frac{a_n}{n^s}} \nonumber$$ 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 $$L_i(k-s)=N_i^s\gamma(s)L_i^*(s) \quad (\forall s\in\textbf{C}), \nonumber$$ $(ii)$ 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 $$L_1(s)=\left(\sum_{u|N}{\frac{a_u}{u^s}}\right)L_2(s) \quad (\forall s\in \textbf{C}). \nonumber$$

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 !

• In your theorem, the two conditions do not directly mention $L_1$ and $L_2$. It would be better to write those conditions using "$L_i$ for $i = 1$ and $2$" in each and writing $L_i$ instead of $L$ in the equations. – KConrad Apr 21 '15 at 19:41
• @Stabilo This kind of questions is usually more appropriate in a direct discussion with someone knowledgeable. I recommend you contact directly someone in your geographical area (when you do, prepare all the details and be professional). – Olivier Apr 21 '15 at 20:58
• I actually went to the fundamental mathematics' lab at my university this afternoon. Unfortunately (or not!), it's the holidays in France and they ends in two weeks. I can not wait! But I know that a direct answer will always be preferable. – Stabilo Apr 21 '15 at 21:09

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 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.