Ramanujan tau function

The Ramanujan tau function, studied by Ramanujan (1916), is the function ${\displaystyle \tau :\mathbb {N} \to \mathbb {Z} }$ defined by the following identity:

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

where ${\displaystyle q=\exp(2\pi iz)}$ with ${\displaystyle \mathrm {Im} (z)>0}$, ${\displaystyle \phi }$ is the Euler function, ${\displaystyle \eta }$ is the Dedekind eta function, and the function ${\displaystyle \Delta (z)}$ is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write ${\displaystyle \Delta /(2\pi )^{12}}$ instead of ${\displaystyle \Delta }$). 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 Dyson (1972).

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

${\displaystyle n}$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
${\displaystyle \tau (n)}$	1	−24	252	−1472	4830	−6048	−16744	84480	−113643	−115920	534612	−370944	−577738	401856	1217160	987136

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.^[1]

Ramanujan (1916) observed, but did not prove, the following three properties of ${\displaystyle \tau (n)}$:

• ${\displaystyle \tau (mn)=\tau (m)\tau (n)}$ if ${\displaystyle \gcd(m,n)=1}$ (meaning that ${\displaystyle \tau (n)}$ is a multiplicative function)
• ${\displaystyle \tau (p^{r+1})=\tau (p)\tau (p^{r})-p^{11}\tau (p^{r-1})}$ for ${\displaystyle p}$ prime and ${\displaystyle r>0}$.
• ${\displaystyle |\tau (p)|\leq 2p^{11/2}}$ for all primes ${\displaystyle p}$.

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 (specifically, he deduced it by applying them to a Kuga-Sato variety).

For ${\displaystyle k\in \mathbb {Z} }$ and ${\displaystyle n\in \mathbb {N} }$, the Divisor function ${\displaystyle \sigma _{k}(n)}$ is the sum of the ${\displaystyle k}$th powers of the divisors of ${\displaystyle n}$. The tau function satisfies several congruence relations; many of them can be expressed in terms of ${\displaystyle \sigma _{k}(n)}$. Here are some:^[2]

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

For ${\displaystyle p\neq 23}$ prime, we have^[2][8]

11. ${\displaystyle \tau (p)\equiv 0\ {\bmod {\ }}23{\text{ if }}\left({\frac {p}{23}}\right)=-1}$
12. ${\displaystyle \tau (p)\equiv \sigma _{11}(p)\ {\bmod {\ }}23^{2}{\text{ if }}p{\text{ is of the form }}a^{2}+23b^{2}}$^[9]
13. ${\displaystyle \tau (p)\equiv -1\ {\bmod {\ }}23{\text{ otherwise}}.}$

In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function:^[10]

${\displaystyle \tau (n)=n^{4}\sigma (n)-24\sum _{i=1}^{n-1}i^{2}(35i^{2}-52in+18n^{2})\sigma (i)\sigma (n-i).}$

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

Suppose that ${\displaystyle f}$ is a weight-${\displaystyle k}$ integer newform and the Fourier coefficients ${\displaystyle a(n)}$ are integers. Consider the problem:

Given that ${\displaystyle f}$ does not have complex multiplication, do almost all primes ${\displaystyle p}$ have the property that ${\displaystyle a(p)\not \equiv 0{\pmod {p}}}$ ?

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 ${\displaystyle a(n){\pmod {p}}}$ for ${\displaystyle n}$ coprime to ${\displaystyle p}$, it is unclear how to compute ${\displaystyle a(p){\pmod {p}}}$. The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes ${\displaystyle p}$ such that ${\displaystyle a(p)=0}$, which thus are congruent to 0 modulo ${\displaystyle p}$. There are no known examples of non-CM ${\displaystyle f}$ with weight greater than 2 for which ${\displaystyle a(p)\not \equiv 0{\pmod {p}}}$ for infinitely many primes ${\displaystyle p}$ (although it should be true for almost all ${\displaystyle p}$. There are also no known examples with ${\displaystyle a(p)\equiv 0{\pmod {p}}}$ for infinitely many ${\displaystyle p}$. Some researchers had begun to doubt whether ${\displaystyle a(p)\equiv 0{\pmod {p}}}$ for infinitely many ${\displaystyle p}$. As evidence, many provided Ramanujan's ${\displaystyle \tau (p)}$ (case of weight 12). The only solutions up to ${\displaystyle 10^{10}}$ to the equation ${\displaystyle \tau (p)\equiv 0{\pmod {p}}}$ are 2, 3, 5, 7, 2411, and 7758337633 (sequence A007659 in the OEIS).^[11]

Lehmer (1947) conjectured that ${\displaystyle \tau (n)\neq 0}$ for all ${\displaystyle n}$, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for ${\displaystyle n}$ up to 214928639999 (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of ${\displaystyle N}$ for which this condition holds for all ${\displaystyle n\leq N}$.

${\displaystyle N}$	reference
3316799	Lehmer (1947)
214928639999	Lehmer (1949)
1000000000000000	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)

Ramanujan's ${\displaystyle L}$-function is defined by

${\displaystyle L(s)=\sum _{n\geq 1}{\frac {\tau (n)}{n^{s}}}}$

if ${\displaystyle \mathrm {Re} (s)>6}$ and by analytic continuation otherwise. It satisfies the functional equation

${\displaystyle {\frac {L(s)\Gamma (s)}{(2\pi )^{s}}}={\frac {L(12-s)\Gamma (12-s)}{(2\pi )^{12-s}}},\quad s\notin \mathbb {Z} _{0}^{-},\,12-s\notin \mathbb {Z} _{0}^{-}}$

and has the Euler product

${\displaystyle L(s)=\prod _{p\,{\text{prime}}}{\frac {1}{1-\tau (p)p^{-s}+p^{11-2s}}},\quad \mathrm {Re} (s)>7.}$

Ramanujan conjectured that all nontrivial zeros of ${\displaystyle L}$ have real part equal to ${\displaystyle 6}$.

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