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Hello,

Let $F$ and $G$ be two functions belonging in the Selberg class, $\Lambda_{F}$ and $\Lambda_{G}$ the complete L-functions associated to $F$ and $G$. I would like to know whether this assertion is true or not:

"$\Lambda_{F}$ and $\Lambda_{G}$ have the same zeroes if and only if $F=G$ or $F=\overline{G}$."

Thank you in advance.

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    $\begingroup$ If $F$ is in the Selberg class, so is $F^2$, $F^3$, etc. So, I think you need to assume $F$ and $G$ are primitive to ask this question. $\endgroup$
    – Micah Milinovich
    Commented Jan 18, 2011 at 22:02
  • $\begingroup$ When I write "the same zeroes", I mean "the same zeroes with the same multiplicity for $\lambda_{F}$ and $\Lambda_{G}$", so that I don't think it's necessary to assume $F$ and $G$ are primitive. $\endgroup$
    – Sylvain JULIEN
    Commented Jan 18, 2011 at 22:15
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    $\begingroup$ Maybe this is a naive comment, but why doesn't your conclusion follow from the Explicit Formula? $\endgroup$
    – Stopple
    Commented Jan 18, 2011 at 22:52
  • $\begingroup$ Major typo: you write F=G or F=G !! This can't be what you intended; please edit! $\endgroup$
    – Zen Harper
    Commented Jan 19, 2011 at 5:02
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    $\begingroup$ Zen, I see no typo: can't you see the overline on $G$ ? $\endgroup$
    – Sylvain JULIEN
    Commented Jan 19, 2011 at 9:46

1 Answer 1

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Expanding on Stopple's comment above, I believe the following argument based on Landau's explicit formula answers the question.

Here is a generalization of Landau's explicit formula for the zeros of the Riemann zeta-function which is exercise 8.4.8 in M. Ram Murty's book Problems in Analytic Number Theory: Let $F$ be in the Selberg class, $n>1$ be a positive integer, and $T>1$. Then $$ \sum_{|\gamma|\leq T} n^\rho = -\frac{T}{\pi}\Lambda_F(n) + O( n^{3/2}\log T )$$ where $\rho=\beta+i\gamma$, $\beta>0$, runs over the non-trivial zeros of $F(s)$. Here the coefficients $\Lambda_F(n)$ are defined by $$ -\frac{F'}{F}(s) = \sum_{n=1} \frac{\Lambda_F(n)}{n^s}.$$

Now suppose that $F$ and $G$ are in the Selberg class and have the same zeros (with multiplicity). Then we deduce from Landau's formula that $$ |\Lambda_F(n) - \Lambda_G(n)| \ll \frac{n^{3/2}\log T}{T} $$ for all $n>1$. Letting $T\rightarrow \infty$, it follows that $\Lambda_F(n) = \Lambda_G(n)$ for all $n>1$. This implies that $F=G$.

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  • $\begingroup$ By the way, has such a generalization of Landau's explicit formula been proven for automorphic L-functions? $\endgroup$
    – Sylvain JULIEN
    Commented Jul 29, 2011 at 19:21

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