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

${\displaystyle \Delta (z)=\sum _{n>0}\tau (n)q^{n}=q\prod _{n>0}\left(1-q^{n}\right)^{24}=q-24q^{2}+252q^{3}-1472q^{4}+4830q^{5}-\cdots ,}$

where ${\displaystyle q=e^{2\pi iz}}$, satisfies

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

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.

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

${\displaystyle L(s,a)=\prod _{p}\left(1+{\frac {a(p)}{p^{s}}}+{\frac {a(p^{2})}{p^{2s}}}+\dots \right)}$

(1)

and due to their completely multiplicative property

${\displaystyle L(s,a)=\prod _{p}\left(1-{\frac {a(p)}{p^{s}}}\right)^{-1}.}$

(2)

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

${\displaystyle L(s,\tau )=\prod _{p}\left(1-{\frac {\tau (p)}{p^{s}}}+{\frac {1}{p^{2s-11}}}\right)^{-1},}$

(3)

where τ(p) is Ramanujan's tau function. The term

${\displaystyle {\frac {1}{p^{2s-11}}}}$

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:

1. τ is multiplicative,
2. 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
3. |τ(p)| ≤ 2p^11/2.

Ramanujan observed that the quadratic equation of u = p^−s in the denominator of RHS of (3),

${\displaystyle 1-\tau (p)u+p^{11}u^{2}}$

would have always imaginary roots from many examples. 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

${\displaystyle \operatorname {Re} (\alpha )=\operatorname {Re} (\beta )=p^{11/2},}$

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:

${\displaystyle O\left(n^{11/2+\varepsilon }\right).}$

In 1917, L. Mordell proved the first two relations using techniques from complex analysis, specifically using 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 (1971). 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.

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 Γ of SL(2, Z). For any modular form

${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}q^{n}\qquad q=e^{2\pi iz},}$

one can form the Dirichlet series

${\displaystyle \varphi (s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}$

For a modular form f (z) of weight k ≥ 2 for Γ, φ(s) absolutely converges in Re(s) > k, because a_n = O(n^k−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:

${\displaystyle R(k-s)=(-1)^{k/2}R(s);}$

this was proved by Wilton in 1929. This correspondence between f and φ is one to one (a₀ = (−1)^k/2 Res_s=k R(s)). Let g(x) = f (ix) −a₀ for x > 0, then g(x) is related with R(s) via the Mellin transformation

${\displaystyle R(s)=\int _{0}^{\infty }g(x)x^{s-1}\,dx\Leftrightarrow g(x)={\frac {1}{2\pi i}}\int _{\operatorname {Re} (s)=\sigma _{0}}R(s)x^{-s}\,ds.}$

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 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 (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 2022) because Deligne's method, which works well in the holomorphic case, does not work in the real analytic case.

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, Howe & 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.

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:

${\displaystyle \pi =\bigotimes \pi _{v}.}$

The Bernstein–Zelevinsky classification tells us that each p-adic π_v can be obtained via unitary parabolic induction from a representation

${\displaystyle \tau _{1,v}\otimes \cdots \otimes \tau _{d,v}.}$

Here each ${\displaystyle \tau _{i,v}}$ is a representation of GL(n_i), over the place v, of the form

${\displaystyle \tau _{i_{0},v}\otimes \left|\det \right|_{v}^{\sigma _{i,v}}}$

with ${\displaystyle \tau _{i_{0},v}}$ tempered. Given n ≥ 2, a Ramanujan bound is a number δ ≥ 0 such that

${\displaystyle \max _{i}\left|\sigma _{i,v}\right|\leq \delta .}$

Langlands classification can be used for the archimedean places. The generalized Ramanujan conjecture is equivalent to the bound δ = 0.

Jacquet, Piatetskii-Shapiro & Shalika (1983) 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.

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.

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 GL(n) implies Selberg's conjecture about eigenvalues of the Laplacian for some discrete groups.

• Blomer, Valentin; Brumley, Farrell (July 2011). "On the Ramanujan conjecture over number fields". Annals of Mathematics. 174 (1): 581–605. arXiv:1003.0559. doi:10.4007/annals.2011.174.1.18. ISSN 0003-486X. MR 2811610. S2CID 54686173.
• Cogdell, J. W.; Kim, H. H.; Piatetski-Shapiro, I. I.; Shahidi, F. (June 2004). "Functoriality for the classical groups". Publications Mathématiques de l'IHÉS. 99 (1): 163–233. CiteSeerX 10.1.1.495.6662. doi:10.1007/s10240-004-0020-z. ISSN 0073-8301. S2CID 7731057.
• Deligne, Pierre (1971). "Formes modulaires et représentations l-adiques". Séminaire Bourbaki. 1968/69: Exposés 347 - 363. Lecture notes in mathematics. Vol. 179. Berlin, New York: Springer-Verlag. doi:10.1007/BFb0058801. ISBN 978-3-540-05356-9.
• Deligne, Pierre (December 1974). "La conjecture de Weil. I". Publications Mathématiques de l'IHÉS (in French). 43 (1): 273–307. doi:10.1007/BF02684373. ISSN 0073-8301. MR 0340258. S2CID 123139343.
• Deligne, Pierre; Serre, Jean-Pierre (1974). "Formes modulaires de poids". Annales Scientifiques de l'École Normale Supérieure. Série 4. 7 (4): 507–530. doi:10.24033/asens.1277. ISSN 0012-9593. MR 0379379.
• Howe, Roger; Piatetski-Shapiro, I. I.; et al. (American Mathematical Society) (1979). "A counterexample to the "generalized Ramanujan conjecture" for (quasi-) split groups". In Borel, Armand; Casselman, Bill (eds.). Automorphic forms, representations, and L-functions. Proceedings of symposia in pure mathematics. Providence, R.I: American Mathematical Society. pp. 315–322. ISBN 978-0-8218-1435-2. MR 0546605.
• Jacquet, H.; Piatetskii-Shapiro, I. I.; Shalika, J. A. (April 1983). "Rankin-Selberg Convolutions". American Journal of Mathematics. 105 (2): 367. doi:10.2307/2374264. JSTOR 2374264. S2CID 124304599.
• Kim, Henry (2002). "Functoriality for the exterior square of 𝐺𝐿₄ and the symmetric fourth of 𝐺𝐿₂" (PDF). Journal of the American Mathematical Society. 16 (1): 139–183. doi:10.1090/S0894-0347-02-00410-1. ISSN 0894-0347.
• Kurokawa, Nobushige (June 1978). "Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two". Inventiones Mathematicae. 49 (2): 149–165. Bibcode:1978InMat..49..149K. doi:10.1007/BF01403084. ISSN 0020-9910. MR 0511188. S2CID 120041528.
• Langlands, R. P. (1970). "Problems in the theory of automorphic forms". In Taam, Choy T. (ed.). Lectures in modern analysis and applications. 3. Lecture notes in mathematics. Vol. 170. Berlin: Springer. pp. 18–61. doi:10.1007/BFb0079065. ISBN 978-3-540-05284-5. MR 0302614.
• Lomelí, L. A. (2009). "Functoriality for the Classical Groups over Function Fields". International Mathematics Research Notices: 4271–4335. doi:10.1093/imrn/rnp089. ISSN 1073-7928. MR 2552304.
• Luo, Wenzhi; Rudnick, Zeév; Sarnak, Peter (1999). Doran, Robert; Dou, Ze-Li; Gilbert, George (eds.). "On the generalized Ramanujan conjecture for 𝐺𝐿(𝑛)". Proc. Sympos. Pure Math. Proceedings of Symposia in Pure Mathematics. 66 (2). Providence, Rhode Island: American Mathematical Society: 301–310. doi:10.1090/pspum/066.2/1703764. ISBN 978-0-8218-1051-4.
• Petersson, Hans (December 1930). "Theorie der automorphen Formen beliebiger reeller Dimension und ihre Darstellung durch eine neue Art Poincaréscher Reihen". Mathematische Annalen (in German). 103 (1): 369–436. doi:10.1007/BF01455702. ISSN 0025-5831. S2CID 122378161.
• Borel, Armand; Casselman, W. (1979). "Multiplicity one theorems". In Borel, Armand; Casselman., W. (eds.). Automorphic forms, representations, and L-functions. Proceedings of Symposia in Pure Mathematics. Providence (Rhode Island): American Mathematical Society. pp. 209–212. ISBN 978-0-8218-1474-1. MR 0546599.
• Ramanujan, Srinivasa (1916). "On certain arithmetical functions" (PDF). Transactions of the Cambridge Philosophical Society. XXII (9): 159–184. Reprinted in Ramanujan Aiyangar, Srinivasa (2000). "Paper 18". In Hardy, Godfrey H. (ed.). Collected papers of Srinivasa Ramanujan (Reprint ed.). Providence, RI: AMS Chelsea Publ. pp. 136–162. ISBN 978-0-8218-2076-6. MR 2280843.
• Sarnak, Peter (2005). "Notes on the generalized Ramanujan conjectures" (PDF). In Clay Mathematics Institute; Arthur, James; Ellwood, D.; Kottwitz, Robert E. (eds.). Harmonic analysis, the trace formula, and Shimura varieties: proceedings of the Clay Mathematics Institute, 2003 Summer School, the Fields Institute, Toronto, Canada, June 2-27, 2003. Clay mathematics proceedings. Vol. 4. Providence, RI: American Mathematical Society. pp. 659–685. ISBN 978-0-8218-3844-0. MR 2192019. OCLC 62282742.
• Satake, Ichirô (1966). "Spherical functions and Ramanujan conjecture". In Borel, Armand; Mostow, George D. (eds.). Algebraic Groups and Discontinuous Subgroups (Boulder, Colo., 1965). Proc. Sympos. Pure Math. Vol. IX. Providence, R.I. pp. 258–264. ISBN 978-0-8218-3213-4. MR 0211955.{{cite book}}: CS1 maint: location missing publisher (link)