Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 137 / 1 / d

Table of contents
- Solution d

Solution

M_{8} (S L_{2} (Z))

is one-dimensional. The two normalized weight-eight forms

E_{8}

and

E_{4}^{2}

therefore agree. Their expansions are

E_{8} = 1 + 480 \sum_{n \geq 1} σ_{7} (n) q^{n}

and

E_{4}^{2} = 1 + 480 \sum_{n \geq 1} σ_{3} (n) q^{n} + 57600 \sum_{n \geq 2} (\sum_{i = 1}^{n - 1} σ_{3} (i) σ_{3} (n - i)) q^{n} .

Equating the coefficient of

q^{n}

and dividing by

480

proves the divisor-sum convolution identity of weights four and eight

σ_{7} (n) = σ_{3} (n) + 120 \sum_{i = 1}^{n - 1} σ_{3} (i) σ_{3} (n - i) .

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