Reason for Gram's Law about Zeta Zeros

[Swamp Thing]

Why does the Zeta function show the tendency called Gram's Law? : https://mathworld.wolfram.com/GramsLaw.html

And why are there "good" and "bad" gram points as defined in https://mathworld.wolfram.com/GramBlock.html ?

 

[suremarc]

I suppose you could phrase Gram's law as the statement that ##\Re(\zeta(\frac{1}{2}+it))## tends to be positive, or that ##Z(t)## is somehow positively correlated with ##\cos\theta(t)##. Although Gram's law only applies when ##\zeta(\frac{1}{2}+it)## is real, it seems to hold elsewhere on the critical line as well. The plot of ##\Re(\zeta(\frac{1}{2}+it))## is mostly positive, whereas the imaginary part appears to fluctuate rapidly:

In particular, ##\int_0^T\cos\theta(t)\cdot Z(t)\,dt## is positive and large, while ##\int_0^T\sin\theta(t)\cdot Z(t)\,dt## is relatively small and might be positive or negative.

Actually, apparently there is a formula ##\sum_{n\leq N} Z(g_{n-1})Z(g_n) \sim -2(\gamma + 1)N##, which I found a few pages into this paper. I plotted the autocorrelation function of ##Z(g_n)## (integrating from 0 to 100,000) using the approximate formula ##g_n=2\pi e^{1+W(\frac{8n+1}{8e})}##:

Spoiler: Plot

Normalized plot of ##f(\tau)=\int_0^{10^5} Z(g(t))Z(g(t+\tau))\,dt##:

I don't really have a full understanding of what I'm doing, but this is all very intriguing to me.

 

[Swamp Thing]

Thank you.. I'm reading now on a mobile device but I'll probably get back later with a question or two.

 

[Swamp Thing]

...
I plotted the autocorrelation function of ##Z(g_n)## (integrating from 0 to 100,000) using the approximate formula ##g_n=2\pi e^{1+W(\frac{8n+1}{8e})}##:
...

What is the "W" in this formula?

 

[suremarc]

What is the "W" in this formula?

That's the Lambert W function. Nothing special, it's just there since ##\theta(t)\sim \frac{t}{2}\log\frac{t}{2\pi}## for ##t## large, and so its inverse function can be approximated using W.