Effective Weak Universality in Short Intervals
Saeree Wananiyakul, Jörn Steuding, Nithi Rungtanapirom
TL;DR
The paper proves effective weak universality for the Riemann zeta-function on short intervals [T, T+H], with H as small as T^{27/82} unconditionally and down to T^{ε} under the Riemann hypothesis. The core method combines an effective multidimensional Ω-result (via finite Euler products with randomized phases), a Fourier-m mollifier L_Q, and a zero-density bound to transfer approximation from a Dirichlet-polynomial model to ζ(s) on a short interval. It yields explicit, effectively computable bounds on the height T in terms of the target derivative data and the interval length, matching prescribed derivatives of log ζ (and, in a variant, of ζ) at a fixed σ_0∈(1/2,1). The results extend Voronin-type universality to the short-interval setting and provide a framework for approximating analytic target functions by ζ(s+iτ) with effective control, highlighting the role of zero-density estimates and conditional assumptions (RH) in tightening interval lengths and constants.
Abstract
We prove an effective universality theorem of the Riemann zeta-function in short intervals $[T,T+H]$ with $T^{\frac{27}{82}}\le H\le T$ by following an effective multidimensional $Ω$-result of Voronin. Furthermore, we also prove the results in short intervals $[T,T+H]$ with $T^ε\le H\le T$ (for any fixed $ε>0$) under the assumption of the Riemann Hypothesis.