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 η3 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→∞. In addition, we prove that all zeros of G{2}n are simple and that G{2}n has infinitely many SL2(ZZ)-inequivalent zeros. We further show that quasimodular forms of maximal depth have no zeros at CM points. In particular, none of the zeros of G{2}n are CM points. Finally, in the special case G2,2, we show that each Ford circle contains exactly two distinct simple zeros.
Related
- Quasimodular Forms, Partitions, and Primes in Intervals
- Law of the Iterated Logarithm for Weighted Sums of Functionals of Long-Memory Gaussian Sequences
- Holomorphic 1-forms without zeros on Kähler manifolds
- Oscillation of partial sums of the Möbius function and zeros of Riemann's zeta function
- The topology of Kähler manifolds and 1-forms without zeros
BibTeX
@misc{kina2026zeros,
title = {{Zeros of Quasimodular Forms Defined by Iterated Sums}},
author = {Katsumi Kina and Gyucheol Shin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12729},
eprint = {2609.12729},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12729},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Zeros of Quasimodular Forms Defined by Iterated Sums
AU - Katsumi Kina
AU - Gyucheol Shin
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.12729
UR - https://arxiv.org/abs/2609.12729
AB - 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 η^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→∞. 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 show that quasimodular forms of maximal depth have no zeros at CM points. In particular, none of the zeros of G_{2}^n are CM points. Finally, in the special case G_2,2, we show that each Ford circle contains exactly two distinct simple zeros.
ER -