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

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.