Report: Statistics of approximations to zeroes of $\zeta$-function via truncated symmetrized Euler products

TL;DR

This paper investigates how well truncated symmetrized Euler products approximate the zeroes of the Riemann zeta function, analyzing the errors and slopes near zeroes through heuristic models and numerical comparisons.

Contribution

It introduces heuristic models for the differences between approximate and actual zeroes of the zeta function and compares these models with numerical data, providing new insights into the approximation errors.

Findings

Errors behave like powers of log, suggesting a specific error growth pattern.

Slope near zeroes behaves logarithmically, enabling numerical formulas.

Models with neighboring zeroes effectively capture scaled differences.

Abstract

We look at approximations $ \zeta_X $ of the $\zeta$-function introduced in Gonek's paper (arXiv:0704.3448). We look at how close the approximate zeroes are to the actual zeroes when (i) X is fixed (Section 1)(ii) X varies like $ t/2\pi $ (Section 3.1). We establish a heuristic for estimating these differences, involving values of $ F_X^\star(t) $ and its near-constant slopes near zeta-zero ordinates $ \gamma $. In Section 3.2 we see the slope around the zeroes behaves logarithmically and we calculate a numerical formula for it. In Section 3.3 and 3.4 we scale the differences with the slopes and compare them with models involving 1 or 2 pairs of neighbouring zeta-zeroes. In Section 3.5, we also look at how often these models capture these scaled differences accurately. In Section 4, we look at our methods from a theoretical standpoint. In Section 5, we look at how close the approximate zeroes are to the actual zeroes when (i) X is fixed (ii) X varies like $ t/2\pi $. The errors seem to behave like powers of log which should be investigated further from a theoretical standpoint.