Tau-function - Citizendia

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

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

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. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Number theory, a branch of Mathematics, a cusp form is a particular kind of Modular form, distinguished in the case of modular forms for the Modular In Mathematics, Weierstrass's elliptic functions are Elliptic functions that take a particularly simple form (cf Jacobi's elliptic functions) they are named

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. Outside number theory the term multiplicative function is usually used for Completely multiplicative functions This article discusses number theoretic multiplicative In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

The first two properties were proved by Mordell in 1917 and the third one was proved by Deligne in 1974. Louis Joel Mordell ( 28 January 1888 - 12 March 1972) was a British mathematician known for pioneering research in Number theory. Pierre René Viscount Deligne (born 3 October 1944 in Brussels) is a Belgian Mathematician.

Congruences for the tau function

For $k\in\mathbb{Z}$ and $n\in\mathbb{Z}_{>0}$ , 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:

$\tau(n)\equiv\sigma_{11}(n)\ \bmod\ 2^{11}\mbox{ for }n\equiv 1\ \bmod\ 8$

$\tau(n)\equiv 1217\sigma_{11}(n)\ \bmod\ 2^{13}\mbox{ for } n\equiv 3\ \bmod\ 8$

$\tau(n)\equiv 1537\sigma_{11}(n)\ \bmod\ 2^{12}\mbox{ for }n\equiv 5\ \bmod\ 8$

$\tau(n)\equiv 705\sigma_{11}(n)\ \bmod\ 2^{14}\mbox{ for }n\equiv 7\ \bmod\ 8$

$\tau(n)\equiv n^{-610}\sigma_{1231}(n)\ \bmod\ 3^{6}\mbox{ for }n\equiv 1\ \bmod\ 3$

$\tau(n)\equiv n^{-610}\sigma_{1231}(n)\ \bmod\ 3^{7}\mbox{ for }n\equiv 2\ \bmod\ 3$

$\tau(n)\equiv n^{-30}\sigma_{71}(n)\ \bmod\ 5^{3}\mbox{ for }n\not\equiv 0\ \bmod\ 5$

$\tau(n)\equiv n\sigma_{9}(n)\ \bmod\ 7\mbox{ for }n\equiv 0,1,2,4\ \bmod\ 7$

$\tau(n)\equiv n\sigma_{9}(n)\ \bmod\ 7^2\mbox{ for }n\equiv 3,5,6\ \bmod\ 7$

$\tau(n)\equiv\sigma_{11}(n)\ \bmod\ 691.$

For $p\not=23$ prime, we have

$\tau(p)\equiv 0\ \bmod\ 23\mbox{ if }\left(\frac{p}{23}\right)=-1$

$\tau(p)\equiv \sigma_{11}(p)\ \bmod\ 23^2\mbox{ if } p\mbox{ is of the form } a^2+23b^2$

$\tau(p)\equiv -1\ \bmod\ 23\mbox{ otherwise}.$

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