Ramanujan's Tau Function

The Ramanujan tau function, studied by , is the function
$\tau : \mathbb\to\mathbb$ defined by the following identity:
:$\sum_\tau(n)q^n=q\prod_\left(1-q^n\right)^ = q\phi(q)^ = \eta(z)^=\Delta(z),$
where $q=\exp(2\pi iz)$ with $\mathrm(z)>0$, $\phi$ is the Euler function, $\eta$ 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 (some authors, notably Apostol, write $\Delta/(2\pi)^$ instead of $\Delta$). It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares. A formula due to Ian G. Macdonald was given in .

Values

The first few values of the tau function are given in the following table :

Calculating this function on an odd square number (i.e. a centered octagonal number) yields an odd number, whereas for any other number the function yields an even number.

Ramanujan's conjectures

observed, but did 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)
* $\tau(p^)=\tau(p)\tau(p^r)-p^\tau(p^)$ for $p$ prime and $r>0$.
* $, \tau(p), \leq 2p^$ for all primes $p$.

The first two properties were proved by and the third one, called the Ramanujan conjecture, was proved by Deligne in 1974 as a consequence of his proof of the Weil conjectures (specifically, he deduced it by applying them to a Kuga-Sato variety).

Congruences for the tau function

For $k\in\mathbb$ and $n\in\mathbb$, the Divisor function $\sigma_k(n)$ is 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 $\sigma_k(n)$. Here are some:Page 4 of
#$\tau(n)\equiv\sigma_(n)\ \bmod\ 2^\textn\equiv 1\ \bmod\ 8$Due to
#$\tau(n)\equiv 1217 \sigma_(n)\ \bmod\ 2^\text n\equiv 3\ \bmod\ 8$
#$\tau(n)\equiv 1537 \sigma_(n)\ \bmod\ 2^\textn\equiv 5\ \bmod\ 8$
#$\tau(n)\equiv 705 \sigma_(n)\ \bmod\ 2^\textn\equiv 7\ \bmod\ 8$
#$\tau(n)\equiv n^\sigma_(n)\ \bmod\ 3^\textn\equiv 1\ \bmod\ 3$Due to
#$\tau(n)\equiv n^\sigma_(n)\ \bmod\ 3^\textn\equiv 2\ \bmod\ 3$
#$\tau(n)\equiv n^\sigma_(n)\ \bmod\ 5^\textn\not\equiv 0\ \bmod\ 5$
#$\tau(n)\equiv n\sigma_(n)\ \bmod\ 7$Due to D. H. Lehmer
#$\tau(n)\equiv n\sigma_(n)\ \bmod\ 7^2\textn\equiv 3,5,6\ \bmod\ 7$
#$\tau(n)\equiv\sigma_(n)\ \bmod\ 691.$

For $p\neq 23$ prime, we have

11. $\tau(p)\equiv 0\ \bmod\ 23\text\left(\frac\right)=-1$
    
12. $\tau(p)\equiv \sigma_(p)\ \bmod\ 23^2\text p\text a^2+23b^2$
    
13. $\tau(p)\equiv -1\ \bmod\ 23\text.$
    

Explicit formula

In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function:

:$\tau(n)=n^4\sigma(n)-24\sum_^i^2(35i^2-52in+18n^2)\sigma(i)\sigma(n-i).$

where $\sigma(n)$ is the sum of the positive divisors of $n$.

Conjectures on the tau function

Suppose that $f$ is a weight-$k$ integer newform and the Fourier coefficients $a(n)$ are integers. Consider the problem:
: Given that $f$ does not have complex multiplication, do almost all primes $p$ have the property that $a(p)\not\equiv 0\pmod$ ?
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 $a(n)\pmod$ for $n$ coprime to $p$, it is unclear how to compute $a(p)\pmod$. The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes $p$ such that $a(p)=0$, which thus are congruent to 0 modulo $p$. There are no known examples of non-CM $f$ with weight greater than 2 for which $a(p)\not\equiv 0\pmod$ for infinitely many primes $p$ (although it should be true for almost all $p$. There are also no known examples with $a(p)\equiv 0 \pmod$ for infinitely many $p$. Some researchers had begun to doubt whether $a(p)\equiv 0 \pmod$ for infinitely many $p$. As evidence, many provided Ramanujan's $\tau(p)$ (case of weight 12). The only solutions up to $10^$ to the equation $\tau(p)\equiv 0\pmod$ are 2, 3, 5, 7, 2411, and .

conjectured that $\tau(n)\neq 0$ for all $n$, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for $n$ up to (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of $N$ for which this condition holds for all $n\leq N$.

Ramanujan's L-function

Ramanujan's $L$-function is defined by
:$L(s)=\sum_\frac$
if $\mathrm(s)>6$ and by analytic continuation otherwise. It satisfies the functional equation
:$\frac=\frac,\quad s\notin\mathbb_0^-, \,12-s\notin\mathbb_0^$
and has the Euler product
:$L(s)=\prod_\frac,\quad \mathrm(s)>7.$
Ramanujan conjectured that all nontrivial zeros of $L$ have real part equal to $6$.

Notes

References

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*{{Citation
, last=Wilton
, first=J. R.
, title=Congruence properties of Ramanujan's function τ(''n'')
, year=1930
, journal=Proceedings of the London Mathematical Society
, volume=31
, pages=1–10
, doi=10.1112/plms/s2-31.1.1

Modular forms
Multiplicative functions
Srinivasa Ramanujan
Zeta and L-functions