$sum\_\{ngeq\; 1\}tau(n)q^n=qprod\_\{ngeq\; 1\}(1-q^n)^\{24\}.$

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

If one substitutes $q=exp(2pi\; iz)$ with $zinmathfrak\{h\}=\{z\; in\; mathbb\{C\}\; :\; Im\; z\; >\; 0\}$ then the function $Delta(z):mathfrak\{h\}tomathbb\{C\}$ defined by

$Delta(z)=sum\_\{ngeq\; 1\}tau(n)q^n$

is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form.

Ramanujan 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 $rinmathbb\{Z\}\_\{>0\}$
• $|tau(p)|\; leq\; 2p^\{11/2\}$ for all primes p.

The first two properties were proved by Mordell in 1917 and the third one was proved by Deligne in 1974.

Congruences for the tau function

For $kinmathbb\{Z\}$ and $ninmathbb\{Z\}\_\{>0\}$, define $sigma\_k(n)$ as the sum of the $k$-th powers of the divisors of $n$. The tau functions satisfies several congruence relations; many of them can be expressed in terms of $sigma\_k(n)$. Here are some:

$tau(n)equivsigma\_\{11\}(n)\; bmod\; 2^\{11\}mbox\{\; for\; \}nequiv\; 1\; bmod\; 8$

$tau(n)equiv\; 1217sigma\_\{11\}(n)\; bmod\; 2^\{13\}mbox\{\; for\; \}\; nequiv\; 3\; bmod\; 8$

$tau(n)equiv\; 1537sigma\_\{11\}(n)\; bmod\; 2^\{12\}mbox\{\; for\; \}nequiv\; 5\; bmod\; 8$

$tau(n)equiv\; 705sigma\_\{11\}(n)\; bmod\; 2^\{14\}mbox\{\; for\; \}nequiv\; 7\; bmod\; 8$

$tau(n)equiv\; n^\{-610\}sigma\_\{1231\}(n)\; bmod\; 3^\{6\}mbox\{\; for\; \}nequiv\; 1\; bmod\; 3$

$tau(n)equiv\; n^\{-610\}sigma\_\{1231\}(n)\; bmod\; 3^\{7\}mbox\{\; for\; \}nequiv\; 2\; bmod\; 3$

$tau(n)equiv\; n^\{-30\}sigma\_\{71\}(n)\; bmod\; 5^\{3\}mbox\{\; for\; \}nnotequiv\; 0\; bmod\; 5$

$tau(n)equiv\; nsigma\_\{9\}(n)\; bmod\; 7mbox\{\; for\; \}nequiv\; 0,1,2,4\; bmod\; 7$

$tau(n)equiv\; nsigma\_\{9\}(n)\; bmod\; 7^2mbox\{\; for\; \}nequiv\; 3,5,6\; bmod\; 7$

$tau(n)equivsigma\_\{11\}(n)\; bmod\; 691.$

For $pnot=23$ prime, we have

$tau(p)equiv\; 0\; bmod\; 23mbox\{\; if\; \}left(frac\{p\}\{23\}right)=-1$

$tau(p)equiv\; sigma\_\{11\}(p)\; bmod\; 23^2mbox\{\; if\; \}\; pmbox\{\; is\; of\; the\; form\; \}\; a^2+23b^2$

$tau(p)equiv\; -1\; bmod\; 23mbox\{\; otherwise\}.$

ta: இராமானுசன் கணிதத்துளிகள்: டௌ-சார்பு