Is this plot of the argument of the Riemann zeta function around ZetaZero(127) correct?

Question

(*Mathematica 8 start*)
Clear[n, k, t, z, FL, NZ]
N[ZetaZero[127]]
NZ[t_] = Arg[Zeta[1/2 + I*t]]/Pi;
Plot[NZ[t], {t, 280, 284}]
Plot[NZ[t], {t, 282.3, 282.6}]

Look at these two graphs of $$\arg(\zeta(1/2+I\cdot t))$$ around the Riemann zeta function zero 0.5 + 282.465 I:

Is it correct that close to $t=282.465$ the graph should make a jump increase of size $2$, and then should continue only slightly downwards, and then should suddenly decrease a step of almost equal to $1$? Should it be that way?

Answer 1

The argument jumps by $\pi$ at each zero because the curve $\zeta(\frac12+it)$ passes right through the origin. (With the principal argument the jump will be upwards or downwards according to whether the curve passes from the lower half-plane to the upper one, or vice versa).

The large jump by $2\pi$ just before the zero is just caused by normalization of the argument to fall within the standard interval $(-\pi,\pi]$. It shows that the curve crosses the negative real axis shortly before the zero at 282.465.

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Note that the almost stable downwards slope of the graph between the jumps is because what you're really plotting is minus the the Riemann-Siegel theta function, plus $\pi$ times the number of critical zeroes below $t$, modulo $2\pi$, and the theta function is quite smooth for such large $t$.