PreprintCC BY 4.0arXiv
Large values of Hardy's Function from the Selberg class
Abstract
Let ZF(t):=F(1/2+it) H(1/2+it)−1/2 denote Hardy's function, where the function F(s) belongs to the Selberg class with polynomial Euler product and satisfies the functional equation F(s)=H(s)F̅(1−s̅). We prove that for every function of degree $1$ and for certain functions of degree $2$ belonging to the Selberg class, one has maxT≤ t≤ 2T ± ZF(t) &≫ exp(CF √( (log T)/(log log T) ) ), where the constant CF depends on F(s) and are explicitly determined.
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BibTeX
@misc{hossain2026large,
title = {{Large values of Hardy's Function from the Selberg class}},
author = {Ahammad Mostafa Hossain and Kamalakshya Mahatab and Akshaa Vatwani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36885},
eprint = {2610.36885},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36885},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Large values of Hardy's Function from the Selberg class
AU - Ahammad Mostafa Hossain
AU - Kamalakshya Mahatab
AU - Akshaa Vatwani
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36885
UR - https://arxiv.org/abs/2610.36885
AB - Let Z_F(t):=F(1/2+it) H(1/2+it)^−1/2 denote Hardy's function, where the function F(s) belongs to the Selberg class with polynomial Euler product and satisfies the functional equation F(s)=H(s)F̅(1−s̅). We prove that for every function of degree $1$ and for certain functions of degree $2$ belonging to the Selberg class, one has max_T≤ t≤ 2T ± Z_F(t) &≫ exp(C_F √( (log T)/(log log T) ) ), where the constant C_F depends on F(s) and are explicitly determined.
ER -