Upper bounds on gaps between zeros of $L$-functions
Tianyu Zhao
TL;DR
This work establishes two unconditional bounds on gaps between consecutive non-trivial zeros of a broad class of $L$-functions by extending the Hall–Hayman and Siegel approaches to general $L$-functions with degree $m$ and analytic conductor $C_L(T)$. The Hall–Hayman-type result yields a zero near height $T$ with $| ildeeta-T| eq0$ bounded by a function of $ rac{ log rac{ log C_L(T)}{ log(m+2)}}{ logigl( rac{ log C_L(T)}{ log(m+2)}igr)- logigl( logigl( rac{ log C_L(T)}{ log(m+2)}igr)igr)}$, while the Siegel-type bound gives a complementary estimate in terms of $ logigl( logigl(C_L(T)^{1/m}igr)igr)$. The work shows each method dominates in different asymptotic regimes and yields concrete corollaries for Dirichlet and Dedekind zeta-functions, including uniform gap results via root discriminants. The combination of explicit analytic bounds (via a Rademacher-Phragmén–Lindelöf framework) and hyperbolic-geometry techniques provides unconditional control over zero spacings in a broad $L$-function setting, with the constants depending on the degree $m$ and the bound $ heta$ in the Euler factors.
Abstract
We prove two unconditional upper bounds on the gaps between ordinates of consecutive non-trivial zeros of a general $L$-function $L(s)$. This extends previous work of Hall and Hayman (2000) on the Riemann zeta-function and work of Siegel (1945) on Dirichlet $L$-functions. Interestingly, we observe that while Hall and Hayman's method gives a sharper estimate when the degree of $L(s)$ is sufficiently small compared to the analytic conductor, Siegel's method does better in the other regime.