# Tau-function

The Ramanujan tau function, studied by Template:Harvs, is the function ${\displaystyle \tau :\mathbb {N} \to \mathbb {Z} }$ defined by the following identity:

${\displaystyle \sum _{n\geq 1}\tau (n)q^{n}=q\prod _{n\geq 1}(1-q^{n})^{24}=\eta (z)^{24}=\Delta (z),}$

## Values

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

 ${\displaystyle n}$ ${\displaystyle \tau (n)}$ 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

Template:Harvtxt observed, but could not prove, the following three properties of ${\displaystyle \tau (n)}$:

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 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]

For p ≠ 23 prime, we have[1][7]

## Conjectures on ${\displaystyle \tau (n)}$

• Suppose that ${\displaystyle f}$ is a weight ${\displaystyle k}$ integer newform and the Fourier coefficients ${\displaystyle a(n)}$ are integers. Consider the problem: If ${\displaystyle f}$ does not have complex multiplication, prove that almost all primes ${\displaystyle p}$ have the property that ${\displaystyle a(p)}$${\displaystyle 0}$ mod ${\displaystyle p}$. Indeed, most primes should have this property, and hence they are called ordinary. Despite the big advances by Deligne and Serre on Galois representations, which determine ${\displaystyle a(n)}$ mod ${\displaystyle p}$ for ${\displaystyle n}$ coprime to ${\displaystyle p}$, we do not have any clue as to how to compute ${\displaystyle a(p)}$ mod ${\displaystyle p}$.The only theorem in this regard is Elkies' famous result for modular elliptic curves, which indeed guarantees that there are infinitely many primes ${\displaystyle p}$ for which ${\displaystyle a(p)}$ = ${\displaystyle 0}$, which in turn is obviously ${\displaystyle 0}$ mod ${\displaystyle p}$. We do not know any examples of non-CM ${\displaystyle f}$ with weight ${\displaystyle >2}$ for which ${\displaystyle a(p)}$${\displaystyle 0}$ mod ${\displaystyle p}$ for infinitely many primes ${\displaystyle p}$ (although it should be true for almost all ${\displaystyle p}$). We also do not know any examples where ${\displaystyle a(p)}$ = ${\displaystyle 0}$ mod ${\displaystyle p}$ for infinitely many ${\displaystyle p}$. Some people had begun to doubt whether ${\displaystyle a(p)}$ = ${\displaystyle 0}$ mod ${\displaystyle p}$ indeed for infinitely many ${\displaystyle p}$. As evidence, many provided Ramanujan's ${\displaystyle \tau (p)}$ (case of weight ${\displaystyle 12}$). The largest known ${\displaystyle p}$ for which ${\displaystyle \tau (p)}$ = ${\displaystyle 0}$ mod ${\displaystyle p}$ is ${\displaystyle p}$ = ${\displaystyle 7758337633}$. The only solutions to the equation ${\displaystyle \tau (p)}$ ${\displaystyle \equiv }$ ${\displaystyle 0}$ mod ${\displaystyle p}$ are ${\displaystyle p}$ = ${\displaystyle 2,3,5,7,2411,}$ and ${\displaystyle 7758337633}$ up to ${\displaystyle 10^{10}}$[9]
• Lehmer (1947) conjectured that ${\displaystyle \tau (n)}$${\displaystyle 0}$ for all ${\displaystyle n}$, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for ${\displaystyle n}$ < ${\displaystyle 214928639999}$ (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of ${\displaystyle n}$ for which this condition holds.
n reference
3316799 Lehmer (1947)
214928639999 Lehmer (1949)
${\displaystyle 10^{15}}$ Serre (1973, p. 98), Serre (1985)
1213229187071998 Jennings (1993)
22689242781695999 Jordan and Kelly (1999)
22798241520242687999 Bosman (2007)

## Notes

1. Page 4 of Template:Harvnb
2. Due to Template:Harvnb
3. Due to Template:Harvnb
4. Due to Lahivi
5. Due to D. H. Lehmer
6. Due to Template:Harvnb
7. Due to Template:Harvnb
8. Due to J.-P. Serre 1968, Section 4.5
9. Due to N. Lygeros and O. Rozier 2010

## References

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