TY - JOUR

T1 - Density of modular forms with transcendental zeros

AU - Choi, Dohoon

AU - Lee, Youngmin

AU - Lim, Subong

N1 - 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.

PY - 2021/8/15

Y1 - 2021/8/15

N2 - 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=0∞af(n)e2πinz of weight k(f), let ϖ(f):=∑n=0rk(f)|af(n)|, where rk(f)=dimCMk(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=0∞Mk,Z(SL2(Z)) (resp. MZtran=∪k=0∞Mk,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]

AB - 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=0∞af(n)e2πinz of weight k(f), let ϖ(f):=∑n=0rk(f)|af(n)|, where rk(f)=dimCMk(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=0∞Mk,Z(SL2(Z)) (resp. MZtran=∪k=0∞Mk,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]

KW - Density

KW - Modular forms

KW - Transcendental zeros

UR - http://www.scopus.com/inward/record.url?scp=85102639076&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2021.125141

DO - 10.1016/j.jmaa.2021.125141

M3 - Article

AN - SCOPUS:85102639076

VL - 500

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

M1 - 125141

ER -