Joint extreme values of the Riemann zeta function at harmonic points
Qiyu Yang, Shengbo Zhao
TL;DR
The paper addresses joint extreme values of the Riemann zeta function at harmonic points by applying the long resonance method to Dirichlet-series representations and truncated Euler products. It proves a refined lower bound on the line $\sigma=1$ that includes a secondary term $\ell(\log_2 T)^{\ell-1}\log_3 T$, extending Levinson-style results to the multi-point setting. In the critical strip $1/2<\sigma<1$, it obtains a processor-defined lower bound with an explicit constant $S(\sigma,\ell)$ and a $\kappa$-dependent exponent, and offers conditional improvements under the Riemann Hypothesis. A short RH-conditional proof of a related result is also provided. Collectively, the work deepens understanding of coordinated extreme behavior of $\zeta(s)$ along arithmetic-harmonic directions and suggests avenues for extending to general $L$-functions.
Abstract
Using the resonance method, we obtain refined estimates for joint extreme values of the Riemann zeta function at harmonic points, improving upon Levinson's 1972 results and providing new insight into the behavior of the Riemann zeta function. Our proof is primarily based on Dirichlet series theory and the truncated Euler product for the Riemann zeta function. As a corollary, we can recover some previously known extreme value results for the zeta function.