The connection between the Weil conjectures and Ramanujan's conjecture

I'm writing an essay about Ramanujan's conjecture and have some questions:

1 How is Ramanujan's conjecture connected with the Weil conjectures?

2 How could Ramanujan's conjecture be assumed true or deduced when Deligne proved the Weil conjecture?

3 How Is Ramanujan's conjecture connected or equivalent to the Riemann Hypothesis?

4 Are there any good articles on these subjects?

Thanks ann regards.

• there are many expositions, which may or may not be suitable for your intended audience; I liked a recent article by Frans Oort – Carlo Beenakker Apr 27 '17 at 10:21
• @CarloBeenakker thanks, I have also found this article and really liked it – user108974 Apr 27 '17 at 10:33
• @CarloBeenakker the article says 'The answer is negative in the strict sense: there is no implication pRH ⇒ RH (and also no argument the reverse way)' what does negative in strict sense mean? – 1.. Apr 27 '17 at 10:34
• the answer refers to the question: "Does this [pRH] provide any progress for the solution of the classical RH?" So "negative in a strict sense" means there is no logical implication, but one cannot exclude that progress has been made in a wider sense. – Carlo Beenakker Apr 27 '17 at 11:31
• There is a book draft by Brian Conrad on this topic and generalizations: math.leidenuniv.nl/~edix/public_html_rennes/brian.ps – Lennart Meier Apr 28 '17 at 7:59

Basically, the coefficients of an holomorphic cusp form are related to the number of points on a certain smooth projective variety over $\mathbb{F}_p$, and the Weil-Riemann hypothesis gives the neccesary error term for the number of points. This is the content of

The Riemann hypothesis over finite fields was of course proved later by Deligne in his "Weil I" paper.

I'm not sure what the best place to learn these things is (I mean the Ramanujan conjecture, conditionally on the Weil conjecture; for the later there are plenty of resources, see here), other than Deligne's paper, but can find some information here on MO,

As for you third question, there seems to be no relationship between Ramanujan-type conjectures and the location of zeros inside the critical strip, whether you are considering the L-funcion attached to a modular form, or any other L-function of arithmetic or geometric origin.

• Thanks, I thought that Ramanujan's tau function also had a critical strip of non-trivial zeroes, or at least it's L-function, but I may misinterpreted that – user108974 Apr 28 '17 at 10:51
• @ziermensch: Yes, the tau function has an $L$-function. When you properly normalize (define) it, it has a functional equation relating $s$ to $1-s$, and the non-trivial zeros lie in the critical strip $0<\Re(s)<1$. On the other hand, the Ramanujan conjecture has not helped us so far to understand these zeros. There are certain properties of $L$-functions where it helps to know that the coefficients satisfy the Ramanujan conjecture, but the Riemann hypothesis lies pretty far from them (as far as we know). – GH from MO May 1 '17 at 7:28
• @GHfromMO okay, Thanks I understand it now! :) – user108974 May 2 '17 at 17:49
• Ramanujan-type results do give the best possible half-plane of absolute convergence of the $L$-function and Euler product corresponding to a modular form. But, in effect, this just gets us to the edge of the critical strip, just as we know that $\zeta(s)$ has no zeros in $\Re(s)>1$. – paul garrett May 4 '17 at 21:03
• @paulgarrett Of course! Thanks for pointing it out. I was thinking only about the critical strip, I'll edit. – Myshkin May 4 '17 at 21:10