Ramanujan 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. 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).

Values of $|\tau(n)|$ for n < 16,000 with logarithmic scale. The blue line picks only the values of n that are multiples of 121.

Values

The first few values of the tau function are given in the following table (sequence A000594 in the 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 did 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)
• $\tau(p^{r + 1}) = \tau(p)\tau(p^r) - p^{11}\tau(p^{r - 1})$ 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 (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 σ_k(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 σ_k(n). Here are some:^[1]

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

For p ≠ 23 prime, we have^[1][7]

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

Conjectures on τ(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 $a(p) \ne 0 \bmod p$. 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 $a(n) \bmod p$ for $n$ coprime to $p$, we do not have any clue as to how to compute $a(p) \bmod p$. 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 $0 \bmod p$. We do not know any examples of non-CM $f$ with weight $>2$ for which $a(p) \ne 0$ 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 $a(p) = 0 \bmod p$ indeed for infinitely many $p$. As evidence, many provided Ramanujan's $\tau(p)$ (case of weight $12$). The largest known $p$ for which $\tau(p) = 0 \bmod p$ is $p = 7758337633$. The only solutions to the equation $\tau(p) \equiv 0 \bmod p$ are $p = 2, 3, 5, 7, 2411,$ and $7758337633$ up to $10^{10}$.^[9]

Lehmer (1947) conjectured that $\tau(n) \ne 0$ 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 for all ${\displaystyle n\leq N}$.

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)
982149821766199295999	Zeng and Yin (2013)
816212624008487344127999	Derickx, van Hoeij, and Zeng (2013)

Notes

1. Page 4 of Swinnerton-Dyer 1973
2. Due to Kolberg 1962
3. Due to Ashworth 1968
4. Due to Lahivi
5. Due to D. H. Lehmer
6. Due to Ramanujan 1916
7. Due to Wilton 1930
8. Due to J.-P. Serre 1968, Section 4.5
9. 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)
• 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 $\tau$ 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
• 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