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.
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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