Equidistribution of Heegner points and the partition function. (English) Zbl 1227.11106
Let \(p(n)\) be the number of partitions of \(n\). Let \(n>0\) such that \(D_n=24n-1\) is square free. The authors prove that there exists an effective constant \(c>0\) such that for all \(\varepsilon>0\) and \(0<b<\frac{1}{12c}\),
\[
\begin{split} p(n)=\frac{1}{(24n-1)}\sum_{{z_Q\in{\Lambda_{D_n}(6)}\atop \text{Im}(z_Q)>1+(24n-1)^{-b}}}\chi_{12}(Q)\left(1-\frac{1}{2\pi\,\text{Im}(z_Q) }\right)e^{-2\pi i z_Q } \\ +O_{\varepsilon}\left(n^{-(\frac{7}{12}-bc)+\varepsilon}\right)+O_{\varepsilon}\left(n^{-(\frac{1}{2}+b)+\varepsilon}\right), n\to\infty. \end{split}
\]
Here \(\chi_{12}\) is the Legendre symbol acting on the set of positive definite, primitive, integral binary quadratic forms \(Q(x,y)=ax^2+bxy+cy^2\) of discriminant \(b^2-4ac=-D_n\) with \(6\mid a\), \(z_Q=\frac{-b+\sqrt{-D_n}}{12a}\), \(\Lambda_{D_n}(6)\) is the set of Heegner points of discriminant \(-D_n\) on \(X_0(6)\). This is used to sharpen the classical bounds of Hardy and Ramanujan, Rademacher, and Lehmer on the error term in Rademacher’s exact formula for \(p(n)\).
Reviewer: Florin Nicolae (Berlin)
MSC:
| 11P82 | Analytic theory of partitions |
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