On the quaternary form \(x^2+xy+7y^2+z^2+zt+7t^2\) (Q2828359)
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scientific article; zbMATH DE number 6643141
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the quaternary form \(x^2+xy+7y^2+z^2+zt+7t^2\) |
scientific article; zbMATH DE number 6643141 |
Statements
25 October 2016
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cubic theta functions
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modular forms
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quaternary form
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representation of integers
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On the quaternary form \(x^2+xy+7y^2+z^2+zt+7t^2\) (English)
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Let \(R(n)\) be a number of the integer representations of \(n\) by the quaternary quadratic form \(x^2+xy+7y^2+z^2+zt+7t^2\). Then NEWLINE\[NEWLINER(n)=\frac{4}{3}\sigma(n)-\frac{16}{3}\sigma(n/3)+16\sigma(n/9)-36\sigma(n/27)+\frac{8}{3}c(n),NEWLINE\]NEWLINE where \(\sigma(n)=\sum_{d|n} d\) and \(c(n)\) is given by NEWLINE\[NEWLINEq\prod_{j=1}^{\infty} (1-q^{3j})^2(1-q^{9j})^2=\sum_{n=0}^{\infty} c(n)q^n.NEWLINE\]
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