Tau function

In mathematics, Ramanujan's tau function is an arithmetic function which may defined in terms of the Delta form by the formal infinite product

$q\prod _{n=1}^{\infty }\left(1-q^{n}\right)^{24}=\sum _{n}\tau (n)q^{n}.\,$

Since Δ is a Hecke eigenform, the tau function is multiplicative, with formal Dirichlet series and Euler product

$\sum _{n}\tau (n)n^{-s}=\prod _{p}\left(1-\tau (p)p^{-s}+p^{11-2s}\right)^{-1}.\,$