## Abstract

For an even positive integer k, let M_{k,Z}(SL_{2}(Z)) be the set of modular forms of weight k on SL_{2}(Z) with integral Fourier coefficients. Let M_{k,Z}^{tran}(SL_{2}(Z)) be the subset of M_{k,Z}(SL_{2}(Z)) consisting of modular forms with only transcendental zeros on the upper half plane H except all elliptic points of SL_{2}(Z). For a modular form f(z)=∑_{n=0}^{∞}a_{f}(n)e^{2πinz} of weight k(f), let ϖ(f):=∑n=0r_{k(f)}|a_{f}(n)|, where r_{k(f)}=dim_{C}M_{k(f),Z}(SL_{2}(Z))⊗C−1. In this paper, we prove that if k=12 or k≥16, then [Formula presented] as X→∞, where α_{k} denotes the sum of the volumes of certain polytopes. Moreover, if we let M_{Z}=∪_{k=0}^{∞}M_{k,Z}(SL_{2}(Z)) (resp. M_{Z}^{tran}=∪_{k=0}^{∞}M_{k,Z}^{tran}(SL_{2}(Z))) and φ is a monotone increasing function on R^{+} such that φ(x+1)−φ(x)≥Cx^{2} for some positive number C, then we prove [Formula presented]

Original language | English |
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Article number | 125141 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 500 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2021 Aug 15 |

### Bibliographical note

Funding Information:The authors appreciate for referee's careful reading and helpful comments. The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1A2C1007517 ). The third author was supported by the National Research Foundation of Korea (NRF) grant (No. NRF-2019R1C1C1009137 ).

Publisher Copyright:

© 2021 Elsevier Inc.

## Keywords

- Density
- Modular forms
- Transcendental zeros

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics