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.