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Monday, October 3, 2016

Entry 36

Given the nome p=eπiτ and τ=n, then the Ramanujan functions Gn and gn are
Gn=21/4p1/24k=1,3,5,(1+pk)gn=21/4p1/24k=1,3,5,(1pk) These are ubiquitous in Ramanujan's Notebooks. Well-known values are G5=ϕ1/4 and G25=ϕ, where ϕ is the golden ratio. However, given the nome's square q=e2πiτ, note also the three Weber modular functions,
f(τ)=q1/48n=1(1+qn1/2)f1(τ)=q1/48n=1(1qn1/2)f2(τ)=2q1/24n=1(1+qn) Notice the similarity of the definitions. Mathematica doesn't have built-in functions for these, but fortunately can be expressed by the more familiar Dedekind eta function η(τ) with τ=n, Gn=21/4f(τ)=21/4η2(τ)η(τ2)η(2τ) gn=21/4f1(τ)=21/4η(τ2)η(τ)The two functions Gn and gn obey(Gngn)8(G8ng8n)=14which is consequence of Weber'sf1(τ)8+f2(τ)8=f(τ)8Ramanujan calculated many explicit values for Gn and gn, one of which is the remarkable G125=3.63352G125φ5/4=1+15(1+22/5φ1/5(545+(φ5)3/2+545(φ5)3/2))2 with the reciprocal golden ratio φ=1+520.61803.

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