Upper Bounds for the Lowest First Zero in Families of Cuspidal Newforms

TL;DR

Under the assumption of the Generalized Riemann Hypothesis, this paper establishes explicit upper bounds for the lowest zero in families of cuspidal newforms, revealing the distribution of zeros near the central point as the level increases.

Contribution

It provides the first explicit bounds on the lowest zero in families of cuspidal newforms and estimates the proportion of forms with zeros close to the central point.

Findings

At least one form has a zero within 1/4 of the average spacing as level tends to infinity.

Established explicit bounds on the lowest zero in these families.

Quantified the percentage of forms with a fixed number of zeros near the central point.

Abstract

Assuming the Generalized Riemann Hypothesis, the non-trivial zeros of $L$-functions lie on the critical line with the real part $1/2$. We find an upper bound of the lowest first zero in families of even cuspidal newforms of prime level tending to infinity. We obtain explicit bounds using the $n$-level densities and results towards the Katz-Sarnak density conjecture. We prove that as the level tends to infinity, there is at least one form with a normalized zero within $1/4$ of the average spacing. We also obtain the first-ever bounds on the percentage of forms in these families with a fixed number of zeros within a small distance near the central point.