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We look at approximations $ \zeta_X $ of the $\zeta$-function introduced inGonek's paper (arXiv:0704.3448). We look at how close the approximate zeroesare 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 thesedifferences, involving values of $ F_X^\star(t) $ and its near-constant slopesnear zeta-zero ordinates $ \gamma $. In Section 3.2 we see the slope around thezeroes behaves logarithmically and we calculate a numerical formula for it. InSection 3.3 and 3.4 we scale the differences with the slopes and compare themwith 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 differencesaccurately. 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 actualzeroes when (i) X is fixed (ii) X varies like $ t/2\pi $. The errors seem tobehave like powers of log which should be investigated further from atheoretical standpoint.

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