On a possible equivalent of Riemann hypothesis 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 ?
 A: 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
