Tau-function

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

$\sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}(1-q^n)^{24} = \eta(z)^{24}=\Delta(z)$ ,

where $q=\exp(2\pi iz)$ with $\Im z > 0$ and $\eta$ is the Dedekind eta function and the function $\Delta(z)$ is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form.

1 Values
2 Ramanujan's conjectures
3 Congruences for the tau function
4 Conjectures on
5 Notes
6 References

Values

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

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

Ramanujan's conjectures

Ramanujan (1916) observed, but could not prove, the following three properties of $\tau(n)$ :

$\tau(mn) = \tau(m)\tau(n)$ if $\gcd(m,n) = 1$ (meaning that $\tau(n)$ is a multiplicative function)
$\tau(p^{r %2B 1}) = \tau(p)\tau(p^r) - p^{11}\tau(p^{r - 1})$ for p prime and r>0.
$|\tau(p)| \leq 2p^{11/2}$ 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.

Congruences for the tau function

For k ∈ Z and n ∈ Z_>0, define σ_k(n) as the sum of the k-th powers of the divisors of n. The tau functions satisfies several congruence relations; many of them can be expressed in terms of σ_k(n). Here are some:^[1]

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

For p ≠ 23 prime, we have^[1]^[7]

$\tau(p)\equiv 0\ \bmod\ 23\mbox{ if }\left(\frac{p}{23}\right)=-1$
$\tau(p)\equiv \sigma_{11}(p)\ \bmod\ 23^2\mbox{ if } p\mbox{ is of the form } a^2%2B23b^2$ ^[8]
$\tau(p)\equiv -1\ \bmod\ 23\mbox{ otherwise}.$

Conjectures on $\tau(n)$

Suppose that $f$ is a weight $k$ integer newform and the Fourier coefficients $a(n)$ are integers. Consider the problem: If $f$ does not have complex multiplication, prove that 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 $p$ , we do not have any clue as to how to compute $a(p)$ mod $p$ .The only theorem in this regard is Elkies' famous result for modular elliptic curves, which indeed guarantees that there are infinitely many primes $p$ for which $a(p)$ = $0$ , which in turn is obviously $0$ mod $p$ . We do not know any examples of non-CM $f$ with weight $>2$ for which $a(p)$ ≠ $0$ mod $p$ for infinitely many primes $p$ (although it should be true for almost all $p$ ). We also do not know any examples where $a(p)$ = $0$ mod $p$ for infinitely many $p$ . Some people had begun to doubt whether $a(p)$ = $0$ mod $p$ indeed for infinitely many $p$ . As evidence, many provided Ramanujan's $\tau(p)$ (case of weight $12$ ). The largest known $p$ for which $\tau(p)$ = $0$ mod $p$ is $p$ = $7758337633$ . The only solutions to the equation $\tau(p)$ $\equiv$ $0$ mod $p$ are $p$ = $2, 3, 5, 7, 2411,$ and $7758337633$ up to $10^{10}$ ^[9]
Lehmer (1947) conjectured that $\tau(n)$ ≠ $0$ for all $n$ , an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for $n$ < $214928639999$ (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of $n$ for which this condition holds.

n	reference
3316799	Lehmer (1947)
214928639999	Lehmer (1949)
$10^{15}$	Serre (1973, p. 98), Serre (1985)
1213229187071998	Jennings (1993)
22689242781695999	Jordan and Kelly (1999)
22798241520242687999	Bosman (2007)

Notes

^ ^a ^b Page 4 of Swinnerton-Dyer 1973
^ ^a ^b ^c ^d Due to Kolberg 1962
^ ^a ^b Due to Ashworth 1968
^ Due to Lahivi
^ ^a ^b Due to D. H. Lehmer
^ Due to Ramanujan 1916
^ Due to Wilton 1930
^ Due to J.-P. Serre 1968, Section 4.5
^ Due to N. Lygeros and O. Rozier 2010

References

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)
Kolberg, O. (1962), "Congruences for Ramanujan's function τ(n)", Arbok Univ. Bergen Mat.-Natur. Ser. (11), MR 0158873
Lygeros, N. (2010), "A New Solution to the Equation τ(p) ≡ 0 (mod p)", Journal of Integer Sequences 13: Article 10.7.4, http://www.cs.uwaterloo.ca/journals/JIS/VOL13/Lygeros/lygeros5.pdf
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, http://www.archive.org/stream/proceedingsofcam1920191721camb#page/n133
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, http://books.google.com/books?id=GJUEAQAAIAAJ
Ramanujan, Srinivasa (1916), "On certain arithmetical functions", Trans. Cambridge Philos. Soc. 22 (9): 159–184, MR 2280861
Serre, J-P. (1968), "Une interprétation relative à la fonction $\tau$ 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, Modular functions of one variable, III, Lecture Notes in Mathematics, 350, pp. 1–55, ISBN 978-3-540-06483-1, MR 0406931, http://www.springerlink.com/content/978-3-540-06483-1
Wilton, J. R. (1930), "Congruence properties of Ramanujan's function τ(n)", Proc. Lond. Math. Soc. 31: 1–10, doi:10.1112/plms/s2-31.1.1