where , satisfies
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 τ(p) 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 conjectured the following:
- τ is multiplicative,
- τ is not completely multiplicative but for prime p and j in N we have: τ(p j+1) = τ(p)τ(p j ) − p11τ(p j−1 ), and
- |τ(p)| ≤ 2p11/2.
Ramanujan observed that the quadratic equation of u = p−s in the denominator of RHS of (3),
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 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 what are now known as Hecke operators. The third statement followed from the proof of the Weil conjectures by Deligne (1974). 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 Deligne (1968). 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 first two proofs of the Ramanujan conjectures to the automorphic L-function of the discrete subgroups Γ of SL(2, Z). For any modular form
one can form the Dirichlet series
For a modular form f (z) of weight k ≥ 2 for Γ, φ(s) absolutely converges in Re(s) > k, because an = O(nk−1+ε). Since f is a modular form of weight k, (s − k)φ(s) turns out to be an entire and R(s) = (2π)−sΓ(s)φ(s) satisfies the functional equation:
this was proved by Wilton in 1929. This correspondence between f and φ is one to one (a0 = (−1)k/2 Ress=k R(s)). Let g(x) = f (ix) −a0 for x > 0, then g(x) is related with R(s) 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 SL(2, Z).
In the case k ≥ 3 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).
Deligne (1971) used 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 (k − 1)/2 where k is the weight of the form. These results also follow from the Weil conjectures, except for the case k = 1, where it is a result of Deligne & Serre (1974).
The Ramanujan–Petersson conjecture for Maass forms is still open (as of 2016) 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
Satake (1966) reformulated the Ramanujan–Petersson conjecture in terms of automorphic representations for GL(2) 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. Kurokawa (1978) and Howe & Piatetski-Shapiro (1979) showed that the conjecture was also false even for some quasi-split and split groups, by constructing automorphic forms for the unitary group U(2, 1) and the symplectic group Sp(4) that are non-tempered almost everywhere, related to the representation θ10.
After the counterexamples were found, Piatetski-Shapiro (1979) 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 GL(n) 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 GL(n), consider a unitary cuspidal automorphic representation:
Here each is a representation of GL(ni), over the place v, of the form
with tempered. Given n ≥ 2, a Ramanujan bound is a number δ ≥ 0 such that
Jacquet, Piatetski-Shapiro & Shalika (1981) obtain a first bound of δ ≤ 1/2 for the general linear group GL(n), known as the trivial bound. 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.
The Ramanujan-Petersson conjecture over global function fields
Drinfeld's proof of the global Langlands correspondence for GL(2) 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 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.
The most celebrated 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 GL(n) implies Selberg's conjecture about eigenvalues of the Laplacian for some discrete groups.
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