Ramanujan conjecture

Ramanujan conjecture

Let ${\textstyle q=e^{2\pi iz}}$. The discriminant modular form is usually defined by$${\displaystyle \Delta (z)=q\prod _{n>0}\left(1-q^{n}\right)^{24}=\eta (z)^{24},}$$where ${\textstyle \eta (z)}$ is the Dedekind eta function. ${\textstyle \Delta (z)}$ is a holomorphic cusp form of weight 12 and level 1. The Ramanujan tau function${\displaystyle \tau (n)}$ is defined for natural numbers by the Fourier series coefficients of this modular form:$${\displaystyle \Delta (z)=\sum _{n=1}^{\infty }\tau (n)q^{n}=q-24q^{2}+252q^{3}-1472q^{4}+4830q^{5}-\cdots .}$$Ramanujan (1916) conjectured the following: