Ramanujan tau function

The Ramanujan tau function, studied by Ramanujan (1916), is the function ${\displaystyle \tau :\mathbb {N} \rightarrow \mathbb {Z} }$ defined by the following identity:

${\displaystyle \sum _{n\geq 1}\tau (n)q^{n}=q\prod _{n\geq 1}\left(1-q^{n}\right)^{24}=q\phi (q)^{24}=\eta (z)^{24}=\Delta (z),}$

where q = exp(2πiz) with Im z > 0, ${\displaystyle \phi }$ is the Euler function, η is the Dedekind eta function, and the function Δ(z) 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).

The first few values of the tau function are given in the following table (sequence A000594 in the OEIS):

Ramanujan (1916) observed, but did not prove, the following three properties of τ(n):

• τ(mn) = τ(m)τ(n) if gcd(m,n) = 1 (meaning that τ(n) is a multiplicative function)
• τ(p^{r + 1}) = τ(p)τ(p^r) − p¹¹ τ(p^{r − 1}) for p prime and r > 0.
• |τ(p)| ≤ 2p^11/2 for all primes p.

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).

For k ∈ ${\displaystyle \mathbb {Z} }$ and n ∈ ${\displaystyle \mathbb {Z} }$_>0, define σ_k(n) as 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 σ_k(n). Here are some:^[1]

1. ${\displaystyle \tau (n)\equiv \sigma _{11}(n)\ {\bmod {\ }}2^{11}{\text{ for }}n\equiv 1\ {\bmod {\ }}8}$^[2]
2. ${\displaystyle \tau (n)\equiv 1217\sigma _{11}(n)\ {\bmod {\ }}2^{13}{\text{ for }}n\equiv 3\ {\bmod {\ }}8}$^[2]
3. ${\displaystyle \tau (n)\equiv 1537\sigma _{11}(n)\ {\bmod {\ }}2^{12}{\text{ for }}n\equiv 5\ {\bmod {\ }}8}$^[2]
4. ${\displaystyle \tau (n)\equiv 705\sigma _{11}(n)\ {\bmod {\ }}2^{14}{\text{ for }}n\equiv 7\ {\bmod {\ }}8}$^[2]
5. ${\displaystyle \tau (n)\equiv n^{-610}\sigma _{1231}(n)\ {\bmod {\ }}3^{6}{\text{ for }}n\equiv 1\ {\bmod {\ }}3}$^[3]
6. ${\displaystyle \tau (n)\equiv n^{-610}\sigma _{1231}(n)\ {\bmod {\ }}3^{7}{\text{ for }}n\equiv 2\ {\bmod {\ }}3}$^[3]
7. ${\displaystyle \tau (n)\equiv n^{-30}\sigma _{71}(n)\ {\bmod {\ }}5^{3}{\text{ for }}n\not \equiv 0\ {\bmod {\ }}5}$^[4]
8. ${\displaystyle \tau (n)\equiv n\sigma _{9}(n)\ {\bmod {\ }}7{\text{ for }}n\equiv 0,1,2,4\ {\bmod {\ }}7}$^[5]
9. ${\displaystyle \tau (n)\equiv n\sigma _{9}(n)\ {\bmod {\ }}7^{2}{\text{ for }}n\equiv 3,5,6\ {\bmod {\ }}7}$^[5]
10. ${\displaystyle \tau (n)\equiv \sigma _{11}(n)\ {\bmod {\ }}691.}$^[6]

For p ≠ 23 prime, we have^[1][7]

11. ${\displaystyle \tau (p)\equiv 0\ {\bmod {\ }}23{\text{ if }}\left({\frac {p}{23}}\right)=-1}$
12. ${\displaystyle \tau (p)\equiv \sigma _{11}(p)\ {\bmod {\ }}23^{2}{\text{ if }}p{\text{ is of the form }}a^{2}+23b^{2}}$^[8]
13. ${\displaystyle \tau (p)\equiv -1\ {\bmod {\ }}23{\text{ otherwise}}.}$

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).^[9]

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.

• 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
• 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. (ed.), Ramanujan revisited (Urbana-Champaign, Ill., 1987), Boston, MA: Academic Press, pp. 245–268, ISBN 978-0-12-058560-1, MR 0938968
• Ramanujan, Srinivasa (1916), "On certain arithmetical functions", Trans. Camb. Philos. Soc., 22 (9): 159–184, MR 2280861
• Serre, J-P. (1968), "Une interprétation des congruences relatives à la fonction ${\displaystyle \tau }$ 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