Ramanujan–Petersson conjecture
In mathematics, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients  of the cusp form  of weight 

where , satisfies

when  is a prime number. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by , is a generalization to other modular forms or automorphic forms.

Ramanujan L-function 
The Riemann zeta function and the Dirichlet L-function satisfy the Euler product,

and due to their completely multiplicative property

Are there L-functions other than the Riemann zeta function and the Dirichlet L-functions satisfying the above relations? Indeed, the L-functions of automorphic forms satisfy the Euler product (1) but they do not satisfy (2) because they do not have the completely multiplicative property. However, Ramanujan discovered that the L-function of the modular discriminant satisfies the modified relation

where  is Ramanujan's tau function. The term

is thought of as the difference from the completely multiplicative property. The above L-function is called Ramanujan's L-function.

Ramanujan conjecture
Ramanujan conjectured the following:
 is multiplicative,
 is not completely multiplicative but for prime  and  in  we have: , and
.

Ramanujan observed that the quadratic equation of  in the denominator of RHS of ,

 

would have always imaginary roots from many examples. The relationship between roots and coefficients of quadratic equations leads the third relation, called Ramanujan's conjecture. Moreover, for the Ramanujan tau function, let the roots of the above quadratic equation be  and , then

which looks like the Riemann Hypothesis. It implies an estimate that is only slightly weaker for all the , namely for any :

In 1917, L. Mordell proved the first two relations using techniques from complex analysis, specifically what are now known as Hecke operators. The third statement followed from the proof of the Weil conjectures by . The formulations required to show that it was a consequence were delicate, and not at all obvious. It was the work of Michio Kuga with contributions also by Mikio Sato, Goro Shimura, and Yasutaka Ihara, followed by . The existence of the connection inspired some of the deep work in the late 1960s when the consequences of the étale cohomology theory were being worked out.

Ramanujan–Petersson conjecture for modular forms
In 1937, Erich Hecke used Hecke operators to generalize the method of Mordell's proof of the first two conjectures to the automorphic L-function of the discrete subgroups  of . For any modular form

one can form the Dirichlet series

For a modular form  of weight  for ,  absolutely converges in , because . Since  is a modular form of weight ,  turns out to be an entire and  satisfies the functional equation:

this was proved by Wilton in 1929. This correspondence between  and  is one to one (). Let  for , then  is related with  via the Mellin transformation

This correspondence relates the Dirichlet series that satisfy the above functional equation with the automorphic form of a discrete subgroup of .

In the case  Hans Petersson introduced a metric on the space of modular forms, called the Petersson metric (also see Weil–Petersson metric). This conjecture was named after him. Under the Petersson metric it is shown that we can define the orthogonality on the space of modular forms as the space of cusp forms and its orthogonal space and they have finite dimensions. Furthermore, we can concretely calculate the dimension of the space of holomorphic modular forms, using the Riemann–Roch theorem (see the dimensions of modular forms).

 used the Eichler–Shimura isomorphism to reduce the Ramanujan conjecture to the Weil conjectures that he later proved.　The more general Ramanujan–Petersson conjecture for holomorphic cusp forms in the theory of elliptic modular forms for congruence subgroups has a similar formulation, with exponent  where  is the weight of the form. These results also follow from the Weil conjectures, except for the case , where it is a result of .

The Ramanujan–Petersson conjecture for Maass forms is still open (as of 2022) because Deligne's method, which works well in the holomorphic case, does not work in the real analytic case.

Ramanujan–Petersson conjecture for automorphic forms
 reformulated the Ramanujan–Petersson conjecture in terms of automorphic representations for  as saying that the local components of automorphic representations lie in the principal series, and suggested this condition as a generalization of the Ramanujan–Petersson conjecture to automorphic forms on other groups. Another way of saying this is that the local components of cusp forms should be tempered. However, several authors found counter-examples for anisotropic groups where the component at infinity was not tempered.  and  showed that the conjecture was also false even for some quasi-split and split groups, by constructing automorphic forms for the unitary group  and the symplectic group  that are non-tempered almost everywhere, related to the representation .

After the counterexamples were found,  suggested that a reformulation of the conjecture should still hold. The current formulation 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 Ramanujan graphs by Lubotzky, Phillips and Sarnak. Indeed, the name "Ramanujan graph" was derived from this connection. Another application is that the Ramanujan–Petersson conjecture for the general linear group  implies Selberg's conjecture about eigenvalues of the Laplacian for some discrete groups.

References

 Reprinted in 

Modular forms
Zeta and L-functions
Srinivasa Ramanujan
Conjectures