Ramanujan conjecture

In the mathematical field of modular forms , the Ramanujan conjecture is an estimate assumed by Ramanujan and proven by Deligne for the Fourier coefficients of the modular discriminants , with applications in graph theory , number theory , representation theory and numerous other areas of mathematics and theoretical computer science. There are also versions for other modular forms (Ramanujan-Petersson conjecture).

Ramanujan's tau function

The Dedekind η function is defined as an infinite product: ${\ displaystyle z \ in {\ mathbb {H}} = \ left \ {z \ in \ mathbb {C} \ mid \ mathrm {Im} \, z> 0 \ right \}}$

{\ displaystyle \ eta (z): = e ^ {\ pi iz / 12} \ prod _ {n = 1} ^ {\ infty} (1-e ^ {2 \ pi inz})}

.

Your 24th power is the discriminant

{\ displaystyle \ Delta (z) = \ eta (z) ^ {24}}

.

With you get ${\ displaystyle q = \ exp (2 \ pi iz)}$

{\ displaystyle \ Delta (z) = q \ prod _ {n \ geq 1} (1-q ^ {n}) ^ {24}}

,

what can be put into a power series in ${\ displaystyle q}$

{\ displaystyle q \ prod _ {n \ geq 1} (1-q ^ {n}) ^ {24} = \ sum _ {n \ geq 1} \ tau (n) q ^ {n}}

whose coefficients (the Fourier coefficients in the q expansion) are the Ramanujan tau function

{\ displaystyle \ tau: \ mathbb {N} \ to \ mathbb {Z}}

(Follow A000594 in OEIS ).

The first values are:

${\ displaystyle n}$	1	2	3	4th	5	6th	7th	8th	9	10	11	12	13	14th	15th	16
${\ displaystyle \ tau (n)}$	1	−24	252	−1472	4830	−6048	−16744	84480	−113643	−115920	534612	−370944	−577738	401856	1217160	987136

Ramanujan discovered many arithmetic properties of the tau function (such as congruences), which then played an important role in the development of the theory of modular forms (for example in the theory of the Hecke operators , where the values of the tau function are eigenvalues of the Hecke operators for the discriminants are).

In 1916 Ramanujan made several conjectures about the tau function, besides the Ramanujan conjecture mentioned below:

${\ displaystyle \ tau (mn) = \ tau (m) \ tau (n)}$ for (i.e. the tau function is a multiplicative function) ${\ displaystyle \ gcd (m, n) = 1}$
${\ displaystyle \ tau (p ^ {r + 1}) = \ tau (p) \ tau (p ^ {r}) - p ^ {11} \ tau (p ^ {r-1})}$ for a prime number p and ${\ displaystyle r> 0}$

These were proven by Louis Mordell in 1917 (using methods of the theory of modular functions that were not available to Ramanujan).

There are also very elegant symmetrical forms for the tau function values, which are related to certain powers of the Dedekind eta function, as Freeman Dyson found in the 1970s, with the powers, as Ian G. Macdonald independently found around the same time, corresponded to the dimensions of finite-dimensional simple Lie algebras. Macdonald established relationships with affine root systems of Lie algebras and classical formulas by Hermann Weyl on root systems and Carl Gustav Jacobi ( Jacobi triple product ).

One of Dyson's formulas is:

{\ displaystyle \ tau (n) = {\ frac {1} {1! \, 2! \, 3! \, 4!}} \ sum \ prod _ {i <j} (u_ {i} -u_ { j})}

the sum of all integers ( ) is , , . ${\ displaystyle u_ {i}}$ ${\ displaystyle i = 1, ..., 5}$ ${\ displaystyle u_ {i} = i \, mod \, 5}$ ${\ displaystyle \ sum _ {i = 1} ^ {5} u_ {i} = 0}$ ${\ displaystyle \ sum _ {i = 1} ^ {5} u_ {i} ^ {2} = 10 \, n}$

Ramanujan conjecture

The Ramanujan conjecture says that for all prime numbers the inequality ${\ displaystyle p}$

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

and more generally for all natural numbers the inequality ${\ displaystyle n}$

{\ displaystyle | \ tau (n) | \ leq d (n) n ^ {\ frac {11} {2}}}

applies, where the number of divisors of denotes. It was proved in 1974 by Pierre Deligne as a consequence of the Weil conjectures that he proved. ${\ displaystyle d (n)}$ ${\ displaystyle n}$

An analogous conjecture for tip shapes (weight k) to congruence subgroups of the module group comes from Hans Petersson (1938) (Ramanujan-Petersson conjecture). As with the discriminant (weight k = 12) the exponent is , only for general k: ${\ displaystyle {\ frac {k-1} {2}}}$

{\ displaystyle | \ tau (p) | \ leq 2p ^ {\ frac {k-1} {2}}}

It was also proven by Deligne via the Weil conjectures. There are also versions for automorphic forms in the Langlands program ( Ilja Pjatetskij-Shapiro and others) and for Maass forms (unproven).

Applications

Construction of Ramanujan graphs : Lubotzky-Philips-Sarnak used the Ramanujan conjecture to prove that certain quotients of the p-adic symmetric space are Ramanujan graphs, i.e. have very good expander properties. ${\ displaystyle PGL (2, \ mathbb {Q} _ {p}) / PGL (2, \ mathbb {Z} _ {p})}$

The Ramanujan conjecture can be reformulated into an estimate of the eigenvalues of Hecke operators .

The Ramanujan conjecture can be reformulated into a statement about the automorphic representation to be associated . ${\ displaystyle \ Delta}$

Trivia

The Ramanujan conjecture was part of the logo of the International Congress of Mathematicians 2010 in Hyderabad.

literature

Alexander Lubotzky : Discrete groups, expanding graphs and invariant measures. With an appendix by Jonathan D. Rogawski. Progress in Mathematics, 125. Birkhäuser Verlag, Basel, 1994. ISBN 3-7643-5075-X
Valentin Blomer , Farrell Brumley : The role of the Ramanujan conjecture in analytic number theory. Bull. Amer. Math. Soc. (NS) 50 (2013), no. 2, 267-320. online (pdf)
Robert Alexander Rankin : Ramanujan's tau-function and its generalizations, in George E. Andrews (Ed.), Ramanujan revisited (Urbana-Champaign, Ill., 1987), Academic Press 1988, pp. 245-268

Web links

Individual evidence

^ Mordell, "On Mr. Ramanujan's empirical expansions of modular functions.", Proceedings of the Cambridge Philosophical Society, Volume 19, 1917, pp. 117-124, archive

↑ Unpublished, see Freeman Dyson, Missed opportunities, Bulletin AMS, Volume 73, September 1972, p. 637. According to Dyson, the formulas listed by Dyson were in part by AOL Atkin (unpublished), the Swedish physicist Winquist, Jacobi, Felix Klein and Robert Fricke and others. The essay dealt with missing opportunities for communication between mathematics and physics, in this case with Dyson himself, who failed to see the connection with Lie algebras .

^ Ian Macdonald, Affine root systems and Dedekind -function, Inventiones Mathematicae, Volume 15, 1972, pp. 91-143, SUB Goettingen . An exception was the dimension d = 26, for which according to Dyson no such explanation exists. ${\ displaystyle \ eta}$

↑ S. Ramanujan: On certain arithmetical functions, Trans. Cambridge Phil. Soc. 22: 159-184 (1916).

^ Pierre Deligne: La conjecture de Weil. I , Inst. Hautes Études Sci. Publ. Math. 43 (1974): 273-307.

↑ Alexander Lubotzky , Ralph Phillips , Peter Sarnak : Ramanujan graphs . Combinatorica 8 (1988) no. 3, 261-277.

[1] Mordell, "On Mr. Ramanujan's empirical expansions of modular functions.", Proceedings of the Cambridge Philosophical Society, Volume 19, 1917, pp. 117-124, archive

[2] Unpublished, see Freeman Dyson, Missed opportunities, Bulletin AMS, Volume 73, September 1972, p. 637. According to Dyson, the formulas listed by Dyson were in part by AOL Atkin (unpublished), the Swedish physicist Winquist, Jacobi, Felix Klein and Robert Fricke and others. The essay dealt with missing opportunities for communication between mathematics and physics, in this case with Dyson himself, who failed to see the connection with Lie algebras .