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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.

Values

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

 n $$\tau(n)$$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 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)
$$\tau(p^{r + 1}) = \tau(p)\tau(p^r) - p^{11}\tau(p^{r - 1})$$ 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 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:[1]

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

For p ≠ 23 prime, we have[1][7]

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

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.

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

Page 4 of Swinnerton-Dyer 1973
Due to Kolberg 1962
Due to Ashworth 1968
Due to Lahivi
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, 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

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