# 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 ?

• In fact Z(t) has a negative local maximum near $t=2$. I think Bombieri speak about $\xi(t)$.
– juan
Sep 4, 2019 at 18:50
• 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 Sep 4, 2019 at 19:37
• 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. Sep 4, 2019 at 20:58
• @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 Sep 4, 2019 at 21:49
• Perhaps the user who downvoted felt the question was something of a fishing expedition. Sep 5, 2019 at 0:06

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.
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.