The question is in the title: if I'm not mistaken, what misses to prove that all automorphic L-functions belong to the Selberg class is a proof of the Ramanujan Conjecture. But the Selberg class is expected to be the natural class of L-functions that satisfy the analogue of the Riemann Hypothesis.

Moreover, Ramanujan Conjecture for automorphic L-functions states that, if $\pi$ is an irreducible unitary cuspidal representation of $GL_{m}(\mathbb{Q}_{\mathbb{A}})$, then for any unramified prime $p$, $\vert\alpha(p,j)\vert=1$. The striking fact is that this conjecture is about complex numbers related to a given automorphic L-function that are supposed to lie on the unit circle, whereas the Grand Riemann Hypothesis is about other complex numbers related to this automorphic L-function that are supposed to lie on a straight line.

But we know that, through homographies, that are rather natural transformations of the Riemann sphere, straights lines and circles are essentially the same objects. Furthermore, I guess that, through Hadamard's factorization theorem, one can expect to have a rather straightforward expression of the non trivial zeroes of a given automorphic L-function in terms of all its $\alpha(p,j)$.

So, would a proof of Ramanujan Conjecture together with other known results about automorphic L-functions imply the Grand Riemann Hypothesis?

Thanks in advance.

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    $\begingroup$ I am sorry but I do not understand the logic behind this question. You say that the L-functions in the Selberg class are expected to satisfy RH and that RC would put the L-functions in question into the Selberg class. But why should this imply they actually satisfy RH. It is only expected after all, and not known for L-functions very well-known to be in the Selberg class either. $\endgroup$ – user9072 May 28 '13 at 17:01

No. GRH lies much deeper than the Ramanujan's conjecture for automorphic forms for $Gl_n$ (which of course is already pretty deep). For example, Langlands functoriality would imply Ramanujan's conjecture (by considering symmetric power of an automorphic form), but it is not expected to imply GRH.

The analogy you observe between GRH and Ramanujan is real, but is nothing more than that: an analogy. For example, in the case of an algebraic automorphic form, then Ramanujan is closely related (and in many cases, is provably implied) by the Riemann Hypothesis for varieties over finite field, proved by Deligne. Hence the similarity of forms you notice between GRH and Ramanujan is the same as the similarity between GRH for variety over finite fields and GRH for Riemann. There is a strong analogy, but no one expects to deduce the case of Riemann simply from the known case proved by Deligne.


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