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