Tau-function

The Ramanujan tau function, studied by Template:Harvs, 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):

$n$	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

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

For p ≠ 23 prime, we have

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

Conjectures on $\tau (n)$

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

Lehmer (1947) conjectured that $\tau (n)$ ≠ $0$ 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)

Notes

References

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