Tau Function (other)
Tau function may refer to: * Tau function (integrable systems), in integrable systems * Ramanujan tau function, giving the Fourier coefficients of the Ramanujan modular form * Divisor function In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includin ...

Tau Function (integrable Systems)
Tau functions are an important ingredient in the modern mathematical theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by :simple:Ryogo Hirota, Ryogo Hirota in his ''direct method'' approach to soliton equations, based on expressing them in an equivalent bilinear form. The term tau function, or \tau -function, was first used systematically by Mikio Sato Sato, Mikio, "Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds", ''Kokyuroku, RIMS, Kyoto Univ.'', 30–46 (1981). and his students in the specific context of the Kadomtsev–Petviashvili equation, Kadomtsev–Petviashvili (or KP) equation and related integrable system, integrable hierarchies. It is a central ingredient in the theory of solitons. In this setting, given any \tau-function satisfying a Hirota-type system of bilinear equations (see below), the corresponding solutions of the equations of the integrable hierarchy a ...
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Ramanujan Tau Function
The Ramanujan tau function, studied by , is the function \tau : \mathbb \rarr\mathbb defined by the following identity: :\sum_\tau(n)q^n=q\prod_\left(1-q^n\right)^ = q\phi(q)^ = \eta(z)^=\Delta(z), where with , \phi is the Euler function, is the Dedekind eta function, and the function is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write \Delta/(2\pi)^ instead of \Delta). It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares. A formula due to Ian G. Macdonald was given in . Values The first few values of the tau function are given in the following table : Ramanujan's conjectures observed, but did not prove, the following three properties of : * if (meaning that is a multiplicative function) * for prime and . * for all primes . The first two properties were proved by and the third one, called the Ramanujan co ...
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