of the generalized Ramanujan conjecture is for a globally generic cuspidal automorphic representation of a connected reductive group, where the generic assumption means that the representation admits a Whittaker model. It states that each local component of such a representation should be tempered. It is an observation due to Langlands that establishing functoriality of symmetric powers of automorphic representations of will give a proof of the Ramanujan–Petersson conjecture. Bounds towards Ramanujan over number fields Obtaining the best possible bounds towards the generalized Ramanujan conjecture in the case of number fields has caught the attention of many mathematicians. Each improvement is considered a milestone in the world of modern number theory. In order to understand the Ramanujan bounds for , consider a unitary cuspidal automorphic representation: The Bernstein–Zelevinsky classification tells us that each p-adic can be obtained via unitary parabolic induction from a representation Here each is a representation of , over the place , of the form with tempered. Given , a Ramanujan bound is a number such that Langlands classification can be used for the archimedean places. The generalized Ramanujan conjecture is equivalent to the bound . obtain a first bound of for the general linear group , known as the trivial bound. An important breakthrough was made by , who currently hold the best general bound of for arbitrary and any number field. In the case of , Kim and Sarnak established the breakthrough bound of when the number field is the field of rational numbers, which is obtained as a consequence of the functoriality result of on the symmetric fourth obtained via the Langlands-Shahidi method. Generalizing the Kim-Sarnak bounds to an arbitrary number field is possible by the results of . For reductive groups other than , the generalized Ramanujan conjecture would follow from principle of Langlands functoriality. An important example are the classical groups, where the best possible bounds were obtained by as a consequence of their Langlands functorial lift. The Ramanujan-Petersson conjecture over global function fields Drinfeld's proof of the global Langlands correspondence for over a global function field leads towards a proof of the Ramanujan–Petersson conjecture. Lafforgue (2002) successfully extended Drinfeld's shtuka technique to the case of in positive characteristic. Via a different technique that extends the Langlands-Shahidi method to include global function fields, proves the Ramanujan conjecture for the classical groups. Applications An application of the Ramanujan conjecture is the explicit construction of