View source for Ramanujan function

{{TEX|done}}{{MSC|11F}}

The function $n \mapsto \tau(n)$, where $\tau(n)$ is the coefficient of $x^n$ ($n \ge 1$) in the expansion of the product
$$
D(x) = x \prod_{m=1}^\infty (1 - x^m)^{24}
$$
as a power series:
$$
D(x) = \sum_{n=1}^\infty \tau(n) x^n \ .
$$
If one puts
$$
\Delta(z) = D(\exp(2\pi i z))
$$
then the Ramanujan function is the $n$-th Fourier coefficient of the cusp form $\Delta(z)$, which was first investigated by S. Ramanujan [[#References|[1]]]. Certain values of the Ramanujan function: $\tau(1) = 1$, $\tau(2) = -24$, $\tau(3) = 252$, $\tau(4) = -1472$, $\tau(5) = 4830$, $\tau(6) = -6048$, $\tau(7) = -16744$, $\tau(30) = 9458784518400$. Ramanujan conjectured (and L.J. Mordell proved) the following properties of the Ramanujan function: it is a [[multiplicative arithmetic function]]
$$
\tau(mn) = \tau(m) \tau(n) \ \text{if}\ (m,n) = 1 \,;
$$
and
$$
\tau(p^{n+1}) = \tau(p^n)\tau(p) - p^{11} \tau(p^{n-1}) \ .
$$

Consequently, the calculation of $\tau(n)$ reduces to calculating $\tau(p)$ when $p$ is prime. It is known that $|\tau(p)|  \le p^{11/2}$ (see [[Ramanujan hypothesis|Ramanujan hypothesis]]). It is known that the Ramanujan function satisfies many congruence relations. For example, Ramanujan knew the congruence
$$
\tau(p) \equiv 1 + p^{11} \pmod{691} \ .
$$

Examples of congruence relations discovered later are:
$$
\tau(n) \equiv \sigma_{11}(n) \pmod{2^{11}} \ \text{if}\ n \equiv 1 \pmod 8
$$
$$
\tau(p) \equiv p + p^10 \pmod{25}
$$
etc.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  S. Ramanujan,   "On certain arithmetical functions"  ''Trans. Cambridge Philos. Soc.'' , '''22'''  (1916)  pp. 159–184</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,   "Une interpretation des congruences relatives à la function $\tau$ de Ramanujan"  ''Sém. Delange–Pisot–Poitou (Théorie des nombres)'' , '''9''' :  14  (1967/68)  pp. 1–17</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  O.M. Fomenko,   "Applications of the theory of modular forms to number theory"  ''J. Soviet Math.'' , '''14''' :  4  (1980)  pp. 1307–1362  ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''15'''  (1977)  pp. 5–91</TD></TR>
</table>



====Comments====
D.H. Lehmer asked whether there exists an $n \in \mathbb{N}$ such that $\tau(n) = 0$, [[#References|[a2]]].  This is still (1990) not known, but one believes that the answer is  "no" . For an elementary introduction to the background of $\Delta(z)$, see [[#References|[a1]]].

The properties mentioned can be combined in the Euler product expansion of the [[formal Dirichlet series]]
$$
\sum_n \tau(n) n^{-s} = \prod_p \left(1 - \tau(p) p^{-s} + p^{11-2s} \right)^{-1} 
$$
which follows from $\Delta$ being a Hecke eigenform of weight 12.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,   "Modular functions and Dirichlet series in number theory" , Springer  (1976)</TD></TR></table>