Analogues of Ramanujan’s 24 squares formula. (English) Zbl 1303.11048
The authors determine a class of formulae analogous to the Ramanujan formula for the number of representations of a positive integer as a sum of 24 squares. For example the number of representations of \(n-1\) as a sum of 16 squares and 8 triangular number is
\[
{1\over 691} \sigma_{11}(n)- {1\over 691} \sigma_{11}\Biggl({n\over 2}\Biggr)+ {690\over 691}\tau(n)+ {42152\over 691} \tau\Biggl({n\over 2}\Biggr)+8192\,\tau\Biggl({n\over 2}\Biggr)+ 25\,\omega(n),
\]
where \(\tau\) is the Ramanujan tau-function and where \(\omega(n)\) is defined through
\[
q^3 \prod^\infty_{n=1} (1- q^n)^8(1- q^{4n})^{16}= \sum^\infty_{n=1} \omega(n)\,q^n.
\]
Reviewer: Meinhard Peters (Münster)
MSC:
11E25 | Sums of squares and representations by other particular quadratic forms |
11F27 | Theta series; Weil representation; theta correspondences |
Keywords:
sums of 24 squares; modular forms; sums of squares and triangular numbers; Eisenstein series; Dedekind eta functionReferences:
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