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

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 ${\displaystyle n\leq N}$ .

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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