The relationship between roots and coefficients of quadratic equations leads to the third relation, called Ramanujan's conjecture.
Moreover, for the Ramanujan tau function, let the roots of the above quadratic equation be α and β, then which looks like the Riemann Hypothesis.
It implies an estimate that is only slightly weaker for all the τ(n), namely for any ε > 0: In 1917, L. Mordell proved the first two relations using techniques from complex analysis, specifically using what are now known as Hecke operators.
It was the work of Michio Kuga with contributions also by Mikio Sato, Goro Shimura, and Yasutaka Ihara, followed by Deligne (1971).
It is an observation due to Langlands that establishing functoriality of symmetric powers of automorphic representations of GL(n) will give a proof of the Ramanujan–Petersson conjecture.
Obtaining the best possible bounds towards the generalized Ramanujan conjecture in the case of number fields has caught the attention of many mathematicians.
An important breakthrough was made by Luo, Rudnick & Sarnak (1999), who currently hold the best general bound of δ ≡ 1/2 − (n2+1)−1 for arbitrary n and any number field.
In the case of GL(2), Kim and Sarnak established the breakthrough bound of δ = 7/64 when the number field is the field of rational numbers, which is obtained as a consequence of the functoriality result of Kim (2002) 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 Blomer & Brumley (2011).
For reductive groups other than GL(n), 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 Cogdell et al. (2004) as a consequence of their Langlands functorial lift.
Lafforgue (2002) successfully extended Drinfeld's shtuka technique to the case of GL(n) in positive characteristic.
Via a different technique that extends the Langlands–Shahidi method to include global function fields, Lomelí (2009) proves the Ramanujan conjecture for the classical groups.