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In [[mathematics]], the '''Ramanujan conjecture''', due to {{harvs|txt|authorlink=Srinivasa Ramanujan|first=Srinivasa |last=Ramanujan|year=1916|loc=p.176}}, states that [[Ramanujan's tau function]] given by the [[Fourier coefficient]]s <math>\tau(n)</math> of the [[cusp form]] <math>\Delta(z)</math> of weight 12
:<math>\Delta(z)=\sum_{n> 0}\tau(n)q^n=q\prod_{n>0}(1-q^n)^{24} = q-24q^2+252q^3+\cdots</math>
(where ''q''=''e''<sup>2π''iz''</sup>) satisfies

:<math>|\tau(p)| \leq 2p^{11/2},</math>

when <math>p</math> is a [[prime number]]. The '''generalized Ramanujan conjecture''' or '''Ramanujan–Petersson conjecture''', introduced by {{harvs|txt|authorlink=Hans Petersson|last=Petersson|year=1930}}, 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]],
:{{NumBlk|:|<math>L(s,a)=\prod_p\biggl(1+\frac{a(p)}{p^s}+\frac{a(p^2)}{p^{2s}}\cdots\biggr)</math>|{{EquationRef|1}}}}
and due to their [[completely multiplicative]] property
:{{NumBlk|:|<math>L(s,a)=\prod_p\biggl(1-\frac{a(p)}{p^s}\biggr)^{-1}.</math>|{{EquationRef|2}}}}
Are there L-functions except for the Riemann zeta function and the Dirichlet L-functions satisfying the above relations? Indeed, the L-functions of automorphic form satisfy the Euler product (1) but they do not satisfy (2) because they do not have completely multiplicative property. However, Ramanujan discovery that the L-functions of automorphic form would satisfy the modified relation,
:{{NumBlk|:|<math>L(s,\tau)=\prod_p\biggl(1-\frac{\tau(p)}{p^s}+\frac{1}{p^{2s-11}}\biggr)^{-1},</math>|{{EquationRef|3}}}}
where <math>\tau(p)</math> is the Ramanujan's tau function. The term +1/(p<sup>2s-11</sup>) in (3) is thought as the difference from the completely multiplicative property.
The above L-function is called '''Ramanujan's L-function'''.

==Ramanujan conjecture==
Ramanujan conjectured the followings:
*1, <math>\tau(n)</math> is [[multiplicative function|multiplicative]],
*2, <math>\tau(p)</math> is not completely multiplicative but for prime p <math>\ \ \ \ \tau(p^{j+1})=\tau(p)\tau(p^j)-p^{11}\tau(p^{j-1})\ (j=1,2,3,\dots),</math> and
*3, <math>|\tau(p)|\le p^{11/2}.</math>

Ramanujan observed that the quadratic equation of u=p<sup>-s</sup> in the denominator of RHS of (3),
: <math>1-\tau(p)u+p^{11}u^2</math>
would have always imaginary roots from a lot of 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
:<math>Re(\alpha)=Re(\beta)=p^{11/2}.</math>
Namely, The real part of the roots is equal to p<sup>11/2</sup>, which looks like the [[Riemann Hypothesis]]. It implies an estimate that is only slightly weaker for all the <math>\tau(n)</math>, namely <math>O(n^{\frac{11}{2}+\varepsilon})</math> for any <math>\varepsilon > 0</math>.

In 1917 [[Louis J. Mordell|L. Mordell]] proved the first two relations using complex analysis, which nowaday, is known as [[Hecke operator]]s. Ramanujan conjecture followed from the proof of the [[Weil conjectures]] by {{harvtxt|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 {{harvtxt|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, using [[Hecke operator]]s [[Erich Hecke]] generalized the method of the first two proofs of the Ramanujan conjectures by Mordell to the [[automorphic L-function]] of the discrete subgroups Γ of SL(2,'''R'''). For a [[modular form]]
:<math>f(z)=\sum^\infty_{n=0}a_nq^n\ \ (q=e^{2\pi iz})</math>
one can get the [[Dirichlet series]]
:<math>\varphi(s)=\sum^\infty_{n=1}a_nn^{-s}.</math>
Then for f(z) in the modular form of Γ with weight k ≥ 2, φ(s) absolutely converges in Re(s) > k due to a<sub>n</sub>=O(n<sup>k-1+ε</sub>). Since f belongs to the modular form with weight k, (s-k)φ(s) turns out to be [[entire function|entire]] and R(s)=(2π)<sup>-s</sup>Γ(s)φ(s) satisfies the '''functional equation''':
:<math>R(k-s)=(-1)^{k/2}R(s),</math>
which proved by Wilton in 1929. This correspondence between f and φ is 1 to 1 (a<sub>0</sub>=(-1)<sup>k/2</sup>Res<sub>s=k</sub>R(s)). Let g(x)=f(ix)-a<sub>0</sub> for x > 0, then g(x) is related with R(s) via the [[Mellin transform|Mellin tranformation]]
:<math>R(s)=\int^\infty_0g(x)x^{s-1}dx\Leftrightarrow g(x)=\frac{1}{2\pi i}\int_{Re_{s=\sigma_0}}R(s)x^{-s}ds.</math>
This correspondence relates the Dirichlet series that satisfy the above functional equation with the automorphic form of a discrete subgroup of SL(2,'''R''').

In the case k ≥ 3 [[Hans Petersson]] introduced the metric on the space of modular forms, called [[Petersson inner product|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 form]]s and its orthogonal space and they have finite dimensions. Furthermore, for the holomorphic modular forms we can concretely calculate the dimension of the space of them due to [[Riemann-Roch theorem]]. (see [[modular form#Consequences|the dimensions of modular forms]])

The more general '''Ramanujan–Petersson conjecture''' for holomorphic cusp forms in the theory of elliptic modular forms for [[congruence subgroup]]s has a similar formulation, with exponent (''k''&nbsp;&minus;&nbsp;1)/2 where ''k'' is the weight of the form. These results also follow from the [[Weil conjectures]], except for the case&nbsp;''k''&nbsp;=&nbsp;1, where it is a result of {{harvtxt|Deligne|Serre|1974}}.

The Ramanujan–Petersson conjecture for [[Maass wave form]]s is still open (as of 2013) because the Deligne's method working well in the holomorphic case does not work in real analytic case.

==Ramanujan–Petersson conjecture for automorphic forms==
{{harvtxt|Satake|1966}} reformulated the Ramanujan–Petersson conjecture in terms of [[automorphic representation]]s for ''GL''<sub>2</sub> 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 group]]s where the component at infinity was not tempered. {{harvtxt|Kurokawa|1978}} and {{harvtxt|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''<sub>2,1</sub> and the  [[symplectic group]] ''Sp''<sub>4</sub> that are non-tempered almost everywhere, related to the representation [[θ10|θ<sub>10</sub>]].

After the counterexamples were found, {{harvtxt|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 [[Robert Langlands|Langlands]] that establishing [[Langlands functoriality|functoriality]] of symmetric powers of automorphic representations of ''GL''<sub>n</sub> 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''<sub>n</sub>, consider a unitary cuspidal [[automorphic representation]] π&nbsp;=&nbsp;⊗' π<sub>v</sub>. The [[Bernstein–Zelevinsky classification]] tells us that each [[p-adic]] <math>\pi_v</math> can be obtained via unitary parabolic induction from a representation <math>\tau_{1,v} \otimes \cdots \otimes \tau_{d,v}</math>. Here each <math>\tau_{i,v}</math> is a representation of ''GL''<sub>n<sub>i</sub></sub>, over the place ''v'', of the form <math>\tau_{i_0,v} \otimes |\det|_v^{\sigma_{i,v}}</math> with <math>\tau_{i_0,v}</math> tempered. Given ''n''&nbsp;≥&nbsp;2, a '''Ramanujan bound''' is a number δ&nbsp;≥&nbsp;0 such that <math>\max_i |\sigma_{i,v}| \leq \delta</math>. [[Langlands classification]] can be used for the [[archimedean valuation|archimedean places]]. The generalized Ramanujan conjecture is equivalent to the bound δ&nbsp;=&nbsp;0.

{{harvtxt|Jacquet|Piatetski-Shapiro|Shalika|1981}} obtain a first bound of δ&nbsp;≤&nbsp;1/2 for the [[general linear group]] ''GL''<sub>n</sub>, known as the trivial bound. An important breakthrough was made by {{harvtxt|Luo|Rudnick|Sarnak|1999}}, who currently hold the best general bound of δ&nbsp;≡&nbsp;1/2 - 1/(''n''<sup>2</sup>+1) for arbitrary ''n'' and any [[number field]]. In the case of ''GL''<sub>2</sub>, Kim and Sarnak established the breakthrough bound of δ&nbsp;=&nbsp;7/64 when the number field is the field of [[rational numbers]], which is obtained as a consequence of the functoriality result of {{harvtxt|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 {{harvtxt|Blomer|Brumley|2011}}.

For [[reductive group]]s other than ''GL''<sub>n</sub>, the generalized Ramanujan conjecture will follow from principle of [[Langlands functoriality]]. An important example are the [[classical groups]], where the best possible bounds were obtained by {{harvtxt|Cogdell|Kim|Piatetski-Shapiro|Shahidi|2004}} as a consequence of their Langlands [[Langlands-Shahidi method|functorial lift]].

==The Ramanujan-Petersson conjecture over global function fields==

[[Vladimir Drinfeld|Drinfeld's]] proof of the global [[Langlands functoriality|Langlands correspondence]] for GL(2) over a [[global function field]] leads towards a proof of the Ramanujan–Petersson conjecture. In a magnificent tour de force, [[Lafforgue's theorem|Lafforgue (2002)]] successfully extended [[Drinfeld module|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, {{harvtxt|Lomelí|2009}} proves the Ramanujan conjecture for the [[classical groups]].

==Applications==
The most celebrated application of the Ramanujan conjecture is the explicit construction of [[Ramanujan graph]]s by [[Alexander Lubotzky|Lubotzky]], Phillips and [[Peter Sarnak|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''<sub>n</sub> implies [[Selberg's conjecture]] about eigenvalues of the Laplacian for some discrete groups.

==References==
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*{{Citation | last1=Cogdell | first1=J. W. | last2=Kim |first2=H. H. | last3=Piatetski-Shapiro | first3=I. I. | last4=Shahidi | first4=F. | title=Functoriality for the classical groups | url=http://www.numdam.org/item?id=PMIHES_2004__99__163_0 | year=2004 | journal=[[Publications Mathématiques de l'IHÉS]] | volume=99 | pages=163–233}}
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Séminaire Bourbaki vol. 1968/69 Exposés 347-363 | url=http://www.numdam.org/item?id=SB_1968-1969__11__139_0 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-05356-9 | doi=10.1007/BFb0058801 | year=1971 | volume=179 | chapter=Formes modulaires et représentations l-adiques }}
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=La conjecture de Weil. I. | url=http://www.numdam.org/item?id=PMIHES_1974__43__273_0 | mr=0340258 | year=1974 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | volume=43 | pages=273–307 | doi=10.1007/BF02684373}}
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Serre | first2=Jean-Pierre | author2-link=Jean-Pierre Serre | title=Formes modulaires de poids 1 | url=http://www.numdam.org/item?id=ASENS_1974_4_7_4_507_0 | mr=0379379 | year=1974 | journal=Annales Scientifiques de l'École Normale Supérieure. Quatrième Série | issn=0012-9593 | volume=7 | pages=507–530}}
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*{{Citation | last1=Kim | first1=H. H. | title=Functoriality for the exterior square of GL(4) and symmetric fourth of GL(2) | year=2002 | journal=Journal of the AMS | volume=16 | pages=139–183}}
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*{{Citation | last1=Langlands | first1=R. P. | title=Lectures in modern analysis and applications, III | url=http://publications.ias.edu/rpl/section/21 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series= Lecture Notes in Math | isbn=978-3-540-05284-5 | doi=10.1007/BFb0079065 | mr=0302614 | year=1970 | volume=170 | chapter=Problems in the theory of automorphic forms | pages=18–61}}
*{{Citation | last1=Lomelí | first1=L. | title=Functoriality for the classical groups over function fields | url=http://imrn.oxfordjournals.org | publisher=[[International Mathematics Research Notices|IMRN]] | doi=10.1093/imrn/rnp089 | mr=2552304 | year=2009 | pages=4271–4335}}
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*{{Citation | last1=Petersson | first1=H. | title=Theorie der automorphen Formen beliebiger reeller Dimension und ihre Darstellung durch eine neue Art Poincaréscher Reihen. | language=German | doi=10.1007/BF01455702 | year=1930 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=103 | issue=1 | pages=369–436}}
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*{{Citation | last1=Ramanujan | first1=Srinivasa | author1-link=Srinivasa Ramanujan | title=On certain arithmetical functions | year=1916 | journal=Transactions of the Cambridge Philosophical Society | volume=XXII | issue=9 | pages=159–184}} Reprinted in {{Citation | last1=Ramanujan | first1=Srinivasa | author1-link=Srinivasa Ramanujan | title=Collected papers of Srinivasa Ramanujan | url=http://books.google.com/books?id=EfnFJHlGo1oC | publisher=AMS Chelsea Publishing, Providence, RI | isbn=978-0-8218-2076-6 | mr=2280843 | year=2000|chapter=Paper 18|pages=136–162}}
*{{Citation | last1=Sarnak | first1=Peter | editor1-last=Arthur | editor1-first=James | editor2-last=Ellwood | editor2-first=David | editor3-last=Kottwitz | editor3-first=Robert | title=Harmonic analysis, the trace formula, and Shimura varieties | url=http://www.claymath.org/publications/Harmonic_Analysis/chapter9.pdf  | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Clay Math. Proc. | isbn=978-0-8218-3844-0  | mr=2192019 | year=2005 | volume=4 | chapter=Notes on the generalized Ramanujan conjectures | pages=659–685}}
*{{Citation | last1=Satake | first1=Ichirô | editor1-last=Borel | editor1-first=Armand | editor1-link=Armand Borel | editor2-last=Mostow | editor2-first=George D. | title=Algebraic Groups and Discontinuous Subgroups (Boulder, Colo., 1965) | location=Providence, R.I. | series=Proc. Sympos. Pure Math. | isbn=978-0-8218-3213-4  | mr=0211955 | year=1966 | volume=IX | chapter=Spherical functions and Ramanujan conjecture | pages=258–264}}

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