Where to search for anomalous local extrema of Z(t) ?

Question

Researchers such as Gourdon, who have computed huge numbers of zeros of zeta on the critical line, often list examples of “Lehmer’s phenomenon”, where a pair of zeros are so close together that the RH is “nearly” false. It would have been false if the extremum of the Riemann-Siegel Z function between the zeros had been a negative maximum or a positive minimum. It's not clear whether their computations would have detected any actual failures of RH resulting from more than one local extremum between zeros. I tried searching for multiple local extrema between pairs of zeros, but it was very slow. Should I confine the search to the rare Gram blocks which violate Rosser's rule? Or is this a futile exercise anyway?

Answer 1

It's not really clear from your question what you are hoping to find. If you are looking for examples of Lehmer's phenomenon, you are looking for pairs of zeros on the critical line which are extremely close (relative to the height $t$). Lots of software packages have these calculations built in; Mathematica's ZetaZero[] for example.

If you are looking instead to disprove the Riemann Hypothesis, you can use the excellent asymptotic formula for the number of zeros to height $T$ in the critical strip, and then count the number of sign changes of the real valued Hardy function $Z(t)$ (which is $\zeta(1/2+it)$ 'with the phase removed'). If these differ, there's a zero in the critical strip off the line, and RH fails.

But when you ask 'Is this a futile exercise anyway', the answer is almost certainly 'yes'. Even experts willing to admit the possibility that the Riemann Hypothesis is false tell us not to expect a counterexample until $\log(\log(t))$ is large. See for example this question.

Answer 2

I suppose that more than one local extremum between zeros of the Riemann-Siegel Z function would imply zeros off the critical line at some of these extrema. So any such cases would show up as a discrepancy in the counts mentioned by @stopple. Searching for multiple local extrema is therefore not an efficient way to disprove RH. In any case the counts of zeros on the critical line and in the critical strip have already been found to agree for very large heights in the critical strip (t < 81,702,130.19 here )