On large gaps between zeros of $L$-functions from branches

André LeClair

Published 2017-04-19Version 1

It is commonly believed that the normalized gaps between consecutive ordinates $t_n$ of the zeros of the Riemann zeta function on the critical line can be arbitrarily large. In particular, drawing on analogies with random matrix theory, it has been conjectured that $$\lambda' ={ lim ~ sup} ~( t_{n+1} - t_n ) \frac{ \log( t_n /2 \pi e)}{2\pi}$$ equals $\infty$. In this article we provide arguments, although not a rigorous proof, that $\lambda'$ is finite. A short argument, conditional on the Riemann Hypothesis, gives $\lambda' \geq 3$. Additional arguments lead us to propose $\lambda'\leq 5$. This proposal is consistent with numerous calculations that place lower bounds on $\lambda'$. We present the generalization of this result to all Dirichlet $L$-functions and those based on cusp forms.