Zeros of Quasimodular Forms Defined by Iterated Sums

Abstract

We study the zeros of the quasimodular forms $G_{\{2{\}}^n}$ defined by iterated sums. We first show that, for every $n>0$, $G_{\{2{\}}^n}$ has exactly $n$ simple zeros on each of the vertical half-lines $\Real(τ)=0$ and $\Real(τ)=1/2$, and that the zeros for consecutive values of $n$ satisfy an interlacing property. The proof is based on an expression of $G_{\{2{\}}^n}$ in terms of the $n$-th derivative of ${\eta }^3$ and on the theory of bell-shaped functions, rather than on Rankin--Swinnerton-Dyer method. We also determine the asymptotic behavior of these zeros as $n\to \infty$. In addition, we prove that all zeros of $G_{\{2{\}}^n}$ are simple and that $G_{\{2{\}}^n}$ has infinitely many $SL_2(\ZZ)$-inequivalent zeros. We further establish a transcendence result for zeros of quasimodular forms of maximal depth, which in particular implies that all zeros of $G_{\{2{\}}^n}$ are transcendental. Finally, in the special case $G_{2,2}$, we show that each Ford circle contains exactly two distinct simple zeros.