Thursday, October 31, 2019
Entry 46
Define dk=η(kτ) with the Dedekind eta function η(τ). Then for Level 16 (d38d4d216)2−2=(d21d8d2d216)(d38d4d216)2+2=(d52d8d21d24d216) Or more simply as a−2=ba+2=c where (a,b,c) are the McKay-Thompson series of class 16B for Monster (A185338, A208603) and obeys (16abc+bca)+2a=abc+bca+acb But one of addends is not a moonshine function so this is not one the 9 dependencies found by Conway et al. However, we also have a−4a=(d2d8)4a+4a=(d34d2d28)4 Adding the two together yields 2(d38d4d216)2=(d2d8)4+(d34d2d28)4 and this is one of the 9 dependencies found by Conway, Norton, and Atkins such that the moonshine functions span a linear space of 172−9=163 dimensions.
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