# Differences of ζ-zeros and patterns mod 4 on log-scale (after Tzimie)

This effect was reported on https://habr.com/ru/articles/812405/. The exposition contains glaring errors, but the effect is real! Here is the plot:

Here I take normalized differences of zeros of ζ-function $$d_n≔(Z_{n+1}-Z_n)\log(|Z|/2π)$$ (after this normalization they are ∼1 in magnitude, with the same argument), multiply them by the mod-4-character i$$^n$$, and plot the running sums. These sums change by about ±1 or ±i, and in the scale of the plot above these differences are small, so the corresponding points “seem to be on a continuous curve” — but a priori one does not expect this curve to be “very special”. Question: is there any explanation for the observed pattern?

Update: I added $$||$$ in $$|Z|$$ above, and explained why the resulting points are expected “to go along a curve” (which might have been “Brownian”!). (BTW, $$|Z|$$ is a shortcut for “something like $$|Z_n|$$ or $$|Z_{n+1}|$$” — and I average them in the script below.)

Myself, a couple of years ago I noticed somewhat-similar oscillations of the average value of $$|ζ'|$$ at zeros when averaged on log-scale — however, the effect discovered by Tzimie seems to be much more pronounced, — and maybe even has a chance to explain what I observed. My gp/PARI code (using 2M zeros calculated with 35 decimal places):

fileno_r = fileopen("zeros6-more-precise.gz","r");
{ global(n=0, BS=4, xds = vector(BS), n_b = 0, x_prev = 0, TOT=vector(3), CLR=vector(3));

while(x = fileread(fileno_r),       \\ invariant: first <= last <= first + vals2 - 1.
x = eval(x);
n++;  n_b++; if(n_b>BS, n_b=1); xds[n_b] = (x-x_prev)*log((x+x_prev)/4/Pi); x_prev = x;
if(n_b == 1 && n>1,           \\ have 4 diffs; "color" the variation: non-const characters: 1 -1 1 -1, 0 1 0 -1, 1 0 -1 0
CLR[1] = xds[2] - xds[3] + xds[4] - xds[1];
CLR[2] = xds[2] - xds[4];
CLR[3] = xds[1] - xds[3];
TOT += CLR;
print(TOT[1], "\t", TOT[2], "\t", TOT[3], "\t", CLR[1], "\t", CLR[2], "\t", CLR[3]);
);
);
}


(I did not try to use characters for different moduli — except 2 — calculated by the script above; “with mod 2” the behavior is less anomalous.)

• BTW, since the gp/PARI reads files “by evaluation”, I recommend checking that the downloaded file “contains numbers only”. Like in zcat FILE.gz | perl -wlne "die if /[^\d.]/" (Use single quotes on *nix.) May 9 at 13:57
• Hmm, maybe this is just “a simple beat” between the variable frequency $\exp(\mathrm{i}πN(t)/2)$ (given by the character mod 4) and “the component $\exp(\mathrm{i}t·\log p^k)/k\sqrt{p^k}$ “of the sum over ζ-zeros”? (Here $N(Z_n)≕n$. Here I use Riemann–Weil summation formula considered as “the FT of quasi-crystals — in the language of Dyson’s “Birds and Frogs”…) May 9 at 15:08
• Then every copy of Bessel’s spirals visible above corresponds to a powr of a prime! May 9 at 15:14