: Mathematics: Ramanujan tau function

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mathematical analysis > complex analysis > modular form > Ramanujan tau function

number > number theory > analytic number theory > modular form > Ramanujan tau function

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... > number theory > geometry of numbers > arithmetic geometry > modular form > Ramanujan tau function

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Ramanujan tau function

The Ramanujan tau function, studied by Ramanujan (1916), is the function ${\\displaystyle \ au :\\mathbb {N} \ ightarrow \\mathbb {Z} }$ defined by the following identity:
${\\displaystyle \\sum _{n\\geq 1}\ au (n)q^{n}=q\\prod _{n\\geq 1}\\left(1-q^{n}\ ight)^{24}=q\\phi (q)^{24}=\\eta (z)^{24}=\\Delta (z),}$
where $q = exp(2 πiz)$ with $Im z > 0$ , ${\\displaystyle \\phi }$ is the Euler function, $η$ is the Dedekind eta function, and the function $Δ(z)$ is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write ${\\displaystyle \\Delta /(2\\pi )^{12}}$ instead of ${\\displaystyle \\Delta }$ ). 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 Dyson (1972).
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Ramanujan_tau_function)

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