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Let $S(t)$ be the deviation of the number of zeros of the Riemann zeta function up to height $t$ from the expectation.

What is the largest observed value of $S(t)$ today?

Here is a quote from a paper of Sarnak in 2004: ``...the greatest observed value of $S(t)$ in the range that it has been computed is about $3.2$."

Do we have new numerical calculations showing a larger value for $S(t)$?

One thing that confuses me is that if, in our numerical calculation, we get at most 3 zeroes more/less than the expected value, how convincing are the numerical verification of the Pair Correlation Conjecture (that the distribution of pair correlation of zeta zeros obeys GUE)? In other words, say if I computed millions of zeros and at worst I got only $3$ zeros off, why shouldn't I conclude that they tend to be rigidly spaced, and the gaps that are getting, very small are just exceptions.

This is especially in light of recent work Tao, Lagarias, and Rodgers that shows that GUE and AGUE are basically the same up to test functions whose Fourier transform supported on $[-1, 1].$

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    $\begingroup$ Odlyzko's computations on the nearest neighbour spacings match the GUE predictions extremely closely. There is a huge difference between GUE and AGUE as far as the nearest neighbour spacings go -- AGUE would have spacings quantised at half-integer values of the usual spacing. One simply does not see that. Large values of $S(t)$ probably grow like $\sqrt{\log T \log \log T}$ --- you are unlikely to see large values. I would write to Jonathan Bober or Ghaith Hiary for the current numerical data. $\endgroup$
    – Lucia
    Commented Nov 11, 2019 at 15:55
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    $\begingroup$ Does the work of Odlyzko's reject the AGUE (say in its mild form, that almost all spacing are half integers)? Given that the Alternative Hypothesis is compatible with what is currently known about the pair correlation of the zeros of zeta. $\endgroup$
    – Farzad Aryan
    Commented Nov 11, 2019 at 17:21
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    $\begingroup$ No, how can it -- it is numerical evidence. But the numerical evidence does not at all indicate the Alternative Hypothesis. The only reason we think about the Alternative Hypothesis is that it is related to the Siegel zero problem. $\endgroup$
    – Lucia
    Commented Nov 11, 2019 at 17:30
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    $\begingroup$ Bober & Hiary, 2016: This resulted in the largest computed value of $Z(t) ≈ 16244.8652$ and $S(t) ≈ 3.3455$. research-information.bristol.ac.uk/files/86475229/zetaComp.pdf $\endgroup$
    – MyNinthAccount
    Commented Nov 12, 2019 at 10:05
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    $\begingroup$ Also, they indicate that Sarnak's 3.2 in 2004 wasn't really known? "Nevertheless, previous to these computations, the largest observed value of $S(t)$ seems to have been $−2.9076$, as reported by Gourdon [8]. Table 2 lists 11 spots where we have found values of $|S(t)| > 3.1$, the largest of which is $S(t) ≈ 3.3455$ for $t ≈ 7.75×10^{27}$. " $\endgroup$
    – MyNinthAccount
    Commented Nov 12, 2019 at 10:07

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