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.



The connection between the Weil conjectures and Ramanujan's conjecture
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


* Pierre Deligne, "[Formes modulaires et représentations l-adiques](http://www.numdam.org/article/SB_1968-1969__11__139_0.pdf)" (1969)


The Riemann hypothesis over finite fields was of course proved later by Deligne in his "[Weil I](http://archive.numdam.org/article/PMIHES_1974__43__273_0.pdf)" 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](https://mathoverflow.net/q/114034)), other than Deligne's paper, but can find some information here on MO,


* [Deligne's proof of Ramanujan's conjecture](https://mathoverflow.net/q/30527)
* [Reference request for a proof of Ramanujan's tau conjecture](https://mathoverflow.net/q/19169)


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.


