The Ramanujan tau function gives the Fourier coefficients at positive indices of the modular discriminant: , with . The Serre derivative and vanishing of weight-fourteen level-one cusp forms give and hence the convolution recurrence
The modular discriminant satisfies , while the Serre derivative gives . Their integral Fourier expansions have and , because . Reducing the two identities gives . The coefficient of in is .
The Fourier expansion of a normalized Eisenstein series and the one-dimensional weight-twelve cusp form space give . The coefficient of , for , reduces modulo to . Since is invertible modulo , this proves the congruence. Integrality follows from and the product .
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The Ramanujan tau function, denoted as \(\tau(n)\), is a function in number theory that arises in the study of modular forms. It is defined for positive integers \(n\) and is deeply connected to the theory of partitions and modular forms. ### Definition The tau function is defined via the coefficients of the q-expansion of the modular discriminant \(\Delta(z)\), which is a specific modular form of weight 12.