This observation is based on the numerical calculation of the exponential sum: $$\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\frac{1}{2}+it}$$ It is known that this sum is related to the famous Riemann–Siegel formula.

Now if we denote this sum as $S_1$, and define sum $S_2$ as follows: $$\sum_{n=1}^{[\sqrt[4]{t/2\pi}]} n^{-\frac{1}{2}+it}$$

Numerical calculation shows that there exists a strong similarity between $|S_1|$ and $|S_2|$, more precisely, it seems that we may have $|S_2|\ll |S_1| \ll |S_2|$.

Here is a plot of $|S_1|$ (blue) and $|S_2|$ (green), with $t$ varies in $[10^4-1000,10^4]$

and another with $t$ varies in $[10^6-1000,10^6]$, more calculations have affirmed this phenomenon.

So there goes **the question**: Does there exist a good mathematical explanation of this phenomenon?