Tau-function Information & Tau-function Links at HealthHaven.com

The Ramanujan tau function is the function $\tau:\mathbb{N}\to\mathbb{Z}$ defined by the following identity:

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

η

is the Dedekind eta function.

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
$τ(n)$	1	−24	252	−1472	4830	−6048	−16744	84480	−113643	−115920	534612	−370944	−577738	401856	1217160	987136

If one substitutes $q = exp(2π i z)$ with $z\in\mathfrak{h}=\{z \in \mathbb{C} : \Im z > 0\}$ then the function $\Delta(z):\mathfrak{h}\to\mathbb{C}$ defined by

$\Delta(z)=\sum_{n\geq 1}\tau(n)q^n$

is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form.

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

$τ(m n) = τ(m)τ(n)$ if $gcd(m, n) = 1$ (meaning that $τ(n)$ is a multiplicative function)
$τ(p r + 1) = τ(p)τ(p r) - p 11 τ(p r - 1)$ for p prime and $r\in\mathbb{Z}_{>0}$
$|\tau(p)| \leq 2p^{11/2}$ for all primes p.

The first two properties were proved by Mordell in 1917 and the third one was proved by Deligne in 1974.

[edit] 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]

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

[edit] 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 Serre 1968, Section 4.5

[edit] References

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
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 $τ$ 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, MR 0406931, ISBN 978-3-540-06483-1, 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

[swd-0] Page 4 of Swinnerton-Dyer 1973

[kolberg-1] Due to Kolberg 1962

[ashworth-2] Due to Ashworth 1968

[3] Due to Lahivi

[Lehmer-4] Due to D. H. Lehmer

[5] Due to Ramanujan 1916

[6] Due to Wilton 1930

[7] Due to Serre 1968, Section 4.5

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