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

## TL;DR

The paper argues that the largest normalized gaps between zeros of the Riemann zeta function are finite, challenging the common belief of their potential to be arbitrarily large, and extends the discussion to other L-functions.

## Contribution

It provides a non-rigorous argument suggesting an upper bound on normalized zero gaps, conditional on the Riemann Hypothesis, and generalizes the result to Dirichlet and cusp form L-functions.

## Key findings

- Proposes an upper bound of 3 for normalized gaps under certain conditions.
- Suggests a possible upper bound of 5 for normalized gaps.
- Extends the analysis to Dirichlet and cusp form L-functions.

## Abstract

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. Conditional on the Riemann Hypothesis, we show that if there are no changes of branch between consecutive zeros then $\lambda' \leq 3$, otherwise $\lambda'$ is allowed to be greater than $3$. 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.