The first two properties were proved by Mordell (1917) and the third one, called the Ramanujan conjecture, was proved by Deligne in 1974 as a consequence of his proof of the Weil conjectures (specifically, he deduced it by applying them to a Kuga-Sato variety).
Congruences for the tau functionedit
For k ∈ and n ∈ >0, the Divisor functionσk(n) is the sum of the kth powers of the divisors of n. The tau function satisfies several congruence relations; many of them can be expressed in terms of σk(n). Here are some:[1]
In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function:[9]
Conjectures on τ(n)edit
Suppose that f is a weight-k integer newform and the Fourier coefficients a(n) are integers. Consider the problem:
Given that f does not have complex multiplication, do almost all primes p have the property that a(p) ≢ 0 (mod p)?
Indeed, most primes should have this property, and hence they are called ordinary. Despite the big advances by Deligne and Serre on Galois representations, which determine a(n) (mod p) for n coprime to p, it is unclear how to compute a(p) (mod p). The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes p such that a(p) = 0, which thus are congruent to 0 modulo p. There are no known examples of non-CM f with weight greater than 2 for which a(p) ≢ 0 (mod p) for infinitely many primes p (although it should be true for almost all p). There are also no known examples with a(p) ≡ 0 (mod p) for infinitely many p. Some researchers had begun to doubt whether a(p) ≡ 0 (mod p) for infinitely many p. As evidence, many provided Ramanujan's τ(p) (case of weight 12). The only solutions up to 1010 to the equation τ(p) ≡ 0 (mod p) are 2, 3, 5, 7, 2411, and 7758337633 (sequence A007659 in the OEIS).[10]
Lehmer (1947) conjectured that τ(n) ≠ 0 for all n, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for n up to 214928639999 (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of N for which this condition holds for all n ≤ N.
^Niebur, Douglas (September 1975). "A formula for Ramanujan's $\tau$-function". Illinois Journal of Mathematics. 19 (3): 448–449. doi:10.1215/ijm/1256050746. ISSN 0019-2082.
^N. Lygeros and O. Rozier (2010). "A new solution for the equation " (PDF). Journal of Integer Sequences. 13: Article 10.7.4.
Referencesedit
Apostol, T. M. (1997), "Modular Functions and Dirichlet Series in Number Theory", New York: Springer-Verlag 2nd Ed.
Ashworth, M. H. (1968), Congruence and identical properties of modular forms (D. Phil. Thesis, Oxford)
Dyson, F. J. (1972), "Missed opportunities", Bull. Amer. Math. Soc., 78 (5): 635–652, doi:10.1090/S0002-9904-1972-12971-9, Zbl 0271.01005
Kolberg, O. (1962), "Congruences for Ramanujan's function τ(n)", Arbok Univ. Bergen Mat.-Natur. Ser. (11), MR 0158873, Zbl 0168.29502
Lehmer, D.H. (1947), "The vanishing of Ramanujan's function τ(n)", Duke Math. J., 14 (2): 429–433, doi:10.1215/s0012-7094-47-01436-1, Zbl 0029.34502
Lygeros, N. (2010), "A New Solution to the Equation τ(p) ≡ 0 (mod p)" (PDF), Journal of Integer Sequences, 13: Article 10.7.4
Newman, M. (1972), A table of τ (p) modulo p, p prime, 3 ≤ p ≤ 16067, National Bureau of Standards
Rankin, Robert A. (1988), "Ramanujan's tau-function and its generalizations", in Andrews, George E. (ed.), Ramanujan revisited (Urbana-Champaign, Ill., 1987), Boston, MA: Academic Press, pp. 245–268, ISBN 978-0-12-058560-1, MR 0938968
Serre, J-P. (1968), "Une interprétation des congruences relatives à la fonction de Ramanujan", Séminaire Delange-Pisot-Poitou, 14
Swinnerton-Dyer, H. P. F. (1973), "On l-adic representations and congruences for coefficients of modular forms", in Kuyk, Willem; Serre, Jean-Pierre (eds.), Modular functions of one variable, III, Lecture Notes in Mathematics, vol. 350, pp. 1–55, doi:10.1007/978-3-540-37802-0, ISBN 978-3-540-06483-1, MR 0406931
Wilton, J. R. (1930), "Congruence properties of Ramanujan's function τ(n)", Proceedings of the London Mathematical Society, 31: 1–10, doi:10.1112/plms/s2-31.1.1