Ramanujan tau function
where with and is the Dedekind eta function and the function is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form. It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares. A formula due to Ian G. Macdonald was given in Dyson (1972).
Ramanujan (1916) observed, but did not prove, the following three properties of :
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 function
For k ∈ Z and n ∈ Z>0, define σk(n) as the sum of the k-th 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:
Conjectures on τ(n)
Suppose that is a weight integer newform and the Fourier coefficients are integers. Consider the problem: If does not have complex multiplication, prove that almost all primes have the property that . 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 for coprime to , we do not have any clue as to how to compute . The only theorem in this regard is Elkies' famous result for modular elliptic curves, which indeed guarantees that there are infinitely many primes for which , which in turn is obviously . We do not know any examples of non-CM with weight for which mod for infinitely many primes (although it should be true for almost all ). We also do not know any examples where mod for infinitely many . Some people had begun to doubt whether indeed for infinitely many . As evidence, many provided Ramanujan's (case of weight ). The largest known for which is . The only solutions to the equation are and up to .
Lehmer (1947) conjectured that for all , an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of for which this condition holds.
|Serre (1973, p. 98), Serre (1985)|
|22689242781695999||Jordan and Kelly (1999)|
|982149821766199295999||Zeng and Yin (2013)|
|816212624008487344127999||Derickx, van Hoeij, and Zeng (2013)|
- 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: 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
- Mordell, Louis J. (1917), "On Mr. Ramanujan's empirical expansions of modular functions.", Proceedings of the Cambridge Philosophical Society, 19: 117–124, JFM 46.0605.01
- 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., Ramanujan revisited (Urbana-Champaign, Ill., 1987), Boston, MA: Academic Press, pp. 245–268, ISBN 978-0-12-058560-1, MR 938968
- Ramanujan, Srinivasa (1916), "On certain arithmetical functions", Trans. Cambridge Philos. Soc., 22 (9): 159–184, MR 2280861
- 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, Modular functions of one variable, III, Lecture Notes in Mathematics, 350, pp. 1–55, 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