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I've read in a Bombieri's paper on official problem statement of Riemann hypothesis for Clay Math institute's millennium problems, a statement and what I understood of it is the following :

The Riemann hypothesis is equivalent to the statement that all local maxima of $ξ(t)$ are positive and all local minima are negative, and it has been suggested that if a counterexample exists then it should be in the neighborhood of unusually large peaks of $|ζ( 1/2 + it)|$.

Is there any current development around this approach ?

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    $\begingroup$ In fact Z(t) has a negative local maximum near $t=2$. I think Bombieri speak about $\xi(t)$. $\endgroup$
    – juan
    Sep 4, 2019 at 18:50
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    $\begingroup$ Check Edwards classical book on RZ; chapter 8 has some of this stuff; the idea is that the derivative of the logarithmic derivative of $Z$ or $\xi$ is positive (between critical zeroes) under RH (for $t \ge 10$ or so in the case of $Z$) and one implication follows immediately $\endgroup$
    – Conrad
    Sep 4, 2019 at 19:37
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    $\begingroup$ It may also be interesting to study $\zeta(x+it)$ as a function of $x$ for a a given $t$. I fiddled a little with that some 15 years ago and the plots obtained on maple at that time seemed to show that the derivative of $\vert\xi(x+it)\vert$ wrt $x$ at $x=1/2$ vanishes whenever $t$ is not the ordinate of a non trivial zero. This would support the simplicity of the zeros. $\endgroup$ Sep 4, 2019 at 20:58
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    $\begingroup$ @A.S.A. I'm not sure exactly what you're asking for, but I recommend the search terms "Lehmer pair" and "de Bruijn-Newman constant". If this is the sort of thing you're asking about, then perhaps the most famous recent advance is the paper by Rodgers and Tao arxiv.org/abs/1801.05914 $\endgroup$ Sep 4, 2019 at 21:49
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    $\begingroup$ Perhaps the user who downvoted felt the question was something of a fishing expedition. $\endgroup$ Sep 5, 2019 at 0:06

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As was mentioned in a comment, RH implies that all local maxima of $\Xi(x)$ are positive and all local minima are negative. The question is whether the converse holds for a general enough class of functions, of which the $\Xi$-function happens to be a member.

Almost certainly, the answer is no. A zero off the line (which actually is a pair of zeros) will cause a positive minimum or negative maximum if those zeros are sufficiently close to the line. Here the distance to the line should be measured on the scale of gaps between the nearby zeros on the line. Thus, the converse would involve proving that all the non-real zeros are very close to the line. That is not known for the Riemann zeta-function, and is not going to be proven in the near future.

To gain some intuition, look at the graph of $(x^2 + a)\cos(x)$ for $a>0$. Only if $a$ is small will there be a positive minimum.

I recently finished a paper on this topic, so I incorporated some details about this question in the final section:

https://arxiv.org/abs/2010.15608

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