Ramanujan function

2020 Mathematics Subject Classification: Primary: 11F [MSN][ZBL]

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 [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). 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

[1]	S. Ramanujan, "On certain arithmetical functions" Trans. Cambridge Philos. Soc. , 22 (1916) pp. 159–184
[2]	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
[3]	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

Comments

It is still (1990) not known whether there exists an $n \in \mathbb{N}$ such that $\tau(n) = 0$. One believes that the answer is "no" . For an elementary introduction to the background of $\Delta(z)$, see [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

[a1]	T.M. Apostol, "Modular functions and Dirichlet series in number theory" , Springer (1976)