Density of modular forms with transcendental zeros

Dohoon Choi, Youngmin Lee, Subong Lim

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

For an even positive integer k, let Mk,Z(SL2(Z)) be the set of modular forms of weight k on SL2(Z) with integral Fourier coefficients. Let Mk,Ztran(SL2(Z)) be the subset of Mk,Z(SL2(Z)) consisting of modular forms with only transcendental zeros on the upper half plane H except all elliptic points of SL2(Z). For a modular form f(z)=∑n=0af(n)e2πinz of weight k(f), let ϖ(f):=∑n=0rk(f)|af(n)|, where rk(f)=dimC⁡Mk(f),Z(SL2(Z))⊗C−1. In this paper, we prove that if k=12 or k≥16, then [Formula presented] as X→∞, where αk denotes the sum of the volumes of certain polytopes. Moreover, if we let MZ=∪k=0Mk,Z(SL2(Z)) (resp. MZtran=∪k=0Mk,Ztran(SL2(Z))) and φ is a monotone increasing function on R+ such that φ(x+1)−φ(x)≥Cx2 for some positive number C, then we prove [Formula presented]

Original languageEnglish
Article number125141
JournalJournal of Mathematical Analysis and Applications
Volume500
Issue number2
DOIs
Publication statusPublished - 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

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