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Assuming an L-function is any element of the intersection $\mathcal{L}$ of the Selberg class $\mathcal{S}$ and the class of automorphic L-functions $\mathcal{A}$, define the notion of Galois class of L-functions to be any subclass of $\mathcal{L}$, containing the constant map equal to $1$ and the Riemann zeta function, and that is closed under both the usual product $\times$ and the Rankin-Selberg convolution $\otimes$. Say an element $F$ in a Galois class of L-functions $\mathcal{G}$ is RS-primitive in $\mathcal{G}$ if for all pair $(G,G')$ of elements of $\mathcal{G}$, one has $F\neq G\otimes G'$.

Is there a way to order a given Galois class of L-functions $\mathcal{G}$ and an analogue of the PNT giving the proportion of $RS$-primitive L-functions in $\mathcal{G}$ not exceeding the $n$-th one? If so, can this be used to prove that a positive proportion of L-functions belong to the maximal Galois class of L-functions denoted by $\mathcal{M}$?

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  • $\begingroup$ Your question doesn't make sense because the PNT isn't magic, it is a consequence of the properties of $\zeta(s)$, and there is no equivalent of $\zeta(s)$ generating all the L-functions. And do you have examples of "Galois class" (the name doesn't make sense because there is a true $Gal(\overline{\Bbb{Q}}/\Bbb{Q})$ action on the L-functions at least those for which the Langlands program works) $\endgroup$
    – reuns
    Jul 9, 2019 at 21:21
  • $\begingroup$ The Galois class generated by $\zeta$ is such an example. $\endgroup$
    – Sylvain Julien
    Jul 9, 2019 at 21:29
  • $\begingroup$ What is it ? ${}{}$ $\endgroup$
    – reuns
    Jul 9, 2019 at 21:31
  • $\begingroup$ Isn't it obvious? The smallest subclass of $\mathcal{L}$ in the sense of inclusion containing $\zeta$ closed under the usual product and the Rankin-selberg convolution. $\endgroup$
    – Sylvain Julien
    Jul 10, 2019 at 5:14
  • $\begingroup$ It is very hard to have a mathematical discussion with you. Make some efforts, define it, really, concretely, with all the necessary details, then tell us what interesting property you think it does have... Or maybe you are expecting I do it and conclude that your question has several problems, as in all your other questions ? You asked a lot of questions about the Selberg class but it seems the only L-function you know is $\zeta$. The Dirichlet L-functions, Artin L-functions, modular forms are a subset of the Selberg class, you are supposed to learn about them and use it to check your ideas $\endgroup$
    – reuns
    Jul 10, 2019 at 18:47

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