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 -