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Thursday, October 31, 2019

Levels 1, 2, 3

Recall our notation for the Dedekind eta function, dm=η(mτ) where we suppress the argument τ for ease of notation. Then we have the following moonshine functions:

Level 1
j1A(τ)=((d1d2)8+28(d2d1)16)3 This is the well-known j-function j(τ). In this form, it is easily seen to be a cube. Its most famous value is j1A(1+1632)=6403203 .

Level 2
j2A(τ)=((d1d2)12+26(d2d1)12)2j2B(τ)=(d1d2)24 Alternatively, j2A(τ)=((d1d2d1/2d4)44(d1/2d4d1d2)4)4 In this form, j2A is seen to be a 4th power. The function doesn't have a name, but is notable for the value j2A(1258)=3964 in Ramanujan's well-known pi formula. Note that,
j4B(τ)=j2A(2τ)j8C(τ)=4j2A(2τ)
Level 3
j3A(τ)=((d1d3)6+33(d3d1)6)2j3B(τ)=(d1d3)12j3C(τ)=3j1A(3τ) Alternatively, j3A(τ)=(d21d23+9d39d1d23)6 In this form, j3A is seen to be also a cube. An example is j3A(1+89/32)=3003 which can also be used in a Ramanujan-type pi formula. Notice that j3C is a cube root.

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