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In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan (1916, p.176), states that Ramanujan's tau function given by the Fourier coefficients \tau(n) of the cusp form \Delta(z) of weight 12

$$\Delta(z)=\sum_{n> 0}\tau(n)q^n=q\prod_{n>0}(1-q^n)^{24} = q-24q^2+252q^3+\cdots$$

(where q=e2πiz) satisfies

$$|\tau(p)| \leq 2p^{11/2},$$

when p is a prime number. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by Petersson (1930), is a generalization to other modular forms or automorphic forms.

Ramanujan conjecture

Ramanujan's conjecture implies an estimate that is only slightly weaker for all the $$\tau(n)$$, namely $$O(n^{\frac{11}{2}+\varepsilon})$$ for any $$\varepsilon > 0.$$

This conjecture of Ramanujan followed from the proof of the Weil conjectures by Deligne (1974). The formulations required to show 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

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 wave forms is still open (as of 2011).
Ramanujan–Petersson conjecture for automorphic forms

Satake (1966) reformulated the Ramanujan–Petersson conjecture in terms of automorphic representations for GL2 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 U2,1 and the symplectic group $${\rm Sp}_4$$ that are non-tempered almost everywhere, related to the representation θ10

Piatetski-Shapiro (1979) suggested that the generalized Ramanujan conjecture should still hold for generic cuspidal automorphic representations of a quasi-split reductive group, where a generic cusp form is roughly one with a Whittaker model. It states that each local component of such a representation should be tempered. Lafforgue's theorem implies that the generalized Ramanujan conjecture is true for the general linear group $${\rm GL}_n$$ over a global function field, by an argument due to Langlands (1970). There are known bounds over a number field. Ramanujan bounds for groups other than $${\rm GL}_n$$ can be obtained as an application of known cases of Langlands functoriality.

The Ramanujan–Petersson conjecture for general linear groups implies Selberg's conjecture about eigenvalues of the Laplacian for some discrete groups. In turn, the Ramanujan–Petersson conjecture for general linear groups follows from the Arthur conjectures.
Applications

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.
References

Deligne, Pierre (1971), "Formes modulaires et représentations l-adiques", Séminaire Bourbaki vol. 1968/69 Exposés 347-363, Lecture Notes in Mathematics, 179, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058801, ISBN 978-3-540-05356-9
Deligne, Pierre (1974), "La conjecture de Weil. I.", Publications Mathématiques de l'IHÉS 43: 273–307, doi:10.1007/BF02684373, ISSN 1618-1913, MR0340258
Deligne, Pierre; Serre, Jean-Pierre (1974), "Formes modulaires de poids 1", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série 7: 507–530, ISSN 0012-9593, MR0379379
Howe, Roger; Piatetski-Shapiro, I. I. (1979), "A counterexample to the "generalized Ramanujan conjecture" for (quasi-) split groups", in Borel, Armand; Casselman, W., Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 315–322, ISBN 978-0-8218-1435-2, MR546605
Kurokawa, Nobushige (1978), "Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two", Inventiones Mathematicae 49 (2): 149–165, doi:10.1007/BF01403084, ISSN 0020-9910, MR511188
Langlands, R. P. (1970), "Problems in the theory of automorphic forms", Lectures in modern analysis and applications, III, Lecture Notes in Math, 170, Berlin, New York: Springer-Verlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 978-3-540-05284-5, MR0302614
Petersson, H. (1930), "Theorie der automorphen Formen beliebiger reeller Dimension und ihre Darstellung durch eine neue Art Poincaréscher Reihen." (in German), Mathematische Annalen 103 (1): 369–436, doi:10.1007/BF01455702, ISSN 0025-5831
Piatetski-Shapiro, I. I. (1979), "Multiplicity one theorems", in Borel, Armand; Casselman., W., Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 209–212, ISBN 978-0-8218-1435-2, MR546599
Ramanujan, Srinivasa (1916), "On certain arithmetical functions", Transactions of the Cambridge Philosophical Society XXII (9): 159–184 Reprinted in Ramanujan, Srinivasa (2000), "Paper 18", Collected papers of Srinivasa Ramanujan, AMS Chelsea Publishing, Providence, RI, pp. 136–162, ISBN 978-0-8218-2076-6, MR2280843
Sarnak, Peter (2005), "Notes on the generalized Ramanujan conjectures", in Arthur, James; Ellwood, David; Kottwitz, Robert, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., 4, Providence, R.I.: American Mathematical Society, pp. 659–685, ISBN 978-0-8218-3844-0, MR2192019
Satake, Ichirô (1966), "Spherical functions and Ramanujan conjecture", in Borel, Armand; Mostow, George D., Algebraic Groups and Discontinuous Subgroups (Boulder, Colo., 1965), Proc. Sympos. Pure Math., IX, Providence, R.I.: American Mathematical Society, pp. 258–264, ISBN 978-0-8218-3213-4, MR0211955

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