One can read in this link: http://link.springer.com/article/10.1007/BF02638380 that if $F$ and $G$ are distinct primitive elements of $\mathcal{S}$, then the transcendence degree of $\mathbb{C}[F,G]$ over $\mathbb{C}$ is two. Denote by $A$ this statement. What could be the easiest to prove statement $B$ such that the conjunction of $A$ and $B$ implies that $\mathcal{S}$ has unique factorization? Thanks in advance.
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asked Feb 16, 2013 at 13:15
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$\begingroup$ Do you mean this mainly as a general algebraic question or for the Selberg class specifically. That is, should this B also be applicable to other semigroups (with suitable restrictions). $\endgroup$– user9072Commented Feb 16, 2013 at 13:36
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$\begingroup$ A meta-question/comment: it could make sense to create a tag 'selberg-class' as there are various questions on the site related to it and to then (over time, not all at once) tag old existing ones with it too. Since many of these questions are by you, your opinion (also perhaps your willingness to do this) on this seems crucial. [I assume you know you can create tags, just by using them.] $\endgroup$– user9072Commented Feb 16, 2013 at 13:47
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$\begingroup$ Indeed this idea of creating a tag dedicated to the Selberg class is a good one. As for your first comment, I'm essentially interested in the Selberg class, and not really in other semigroups. $\endgroup$– Sylvain JULIENCommented Feb 16, 2013 at 14:15
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$\begingroup$ Why should algebraic independence be related to unique factorization? $\endgroup$– Marc PalmCommented Feb 16, 2013 at 18:26
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$\begingroup$ Jerzy Jaczorowski, Giuseppe Molteni and Alberto Perelli proved in 1999 that unique factorization in $\mathcal{S}$ is equivalent to algebraic independence of distinct primitive members of $\mathcal{S}$ over the ring of $p$-finite Dirichlet series, which I guess is a conjectured generalization of the algebraic independence mentionned in the link above. Hence my question. $\endgroup$– Sylvain JULIENCommented Feb 16, 2013 at 19:53
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