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.

File:Absolute Tau function for x up to 16,000 with logarithmic scale.JPG

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 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)
• $\tau(p^{r + 1}) = \tau(p)\tau(p^r) - p^{11}\tau(p^{r - 1})$ 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 function satisfies several congruence relations; many of them can be expressed in terms of σ_k(n). Here are some:

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

For p ≠ 23 prime, we have

11. $\tau(p)\equiv 0\ \bmod\ 23\text{ if }\left(\frac{p}{23}\right)=-1$
12. $\tau(p)\equiv \sigma_{11}(p)\ \bmod\ 23^2\text{ if } p\text{ is of the form } a^2+23b^2$
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 $10^{10}$.

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)

References

• 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