Year: 2026
Type: preprint
Abstract: We prove new large value estimates for the Riemann zeta function on the critical line. For instance, we improve the upper bound on the measure of $t\in [T, 2T]$ with $|ζ(1/2 + it)| \geq T^{1/8}$ for the first time since Hardy-Littlewood (1923). Our results imply improved upper bounds on the $k$-th moment of zeta for every $4 < k \leq 12$. We show in particular that \begin{equation*} \int_{0}^{T} |ζ(\tfrac{1}{2} + it)|^6 \,d t \ll_\varepsilon T^{\frac{5}{4} - \frac{1}{60} + \varepsilon}. \end{equation*} The main ingredient is a new large value estimate for exponential sums with square-root phases, and certain perturbations. Those arise in the expansion of the short second moment of zeta.
Source: arXiv (Cornell University)
Authors: de Faveri, Alexandre, Pandey, Mayank
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215832989