On an observation which relates to the exponential sum $\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\frac{1}{2}+it}$ 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?
 A: The assertion is generally believed to be true. In fact, much more is conjectured: For every $\epsilon>0$ we have $\left|\zeta(\frac{1}{2}+it)-\sum_{n\leq t^\epsilon} n^{-\frac{1}{2}-it}\right|= \mathcal{O}(t^\epsilon)$. This statement is equivalent to the Lindelöff hypothesis. One direction is obvious: If $\zeta$ can be approximated by short sums, then $\zeta$ cannot be too large. The other direction can be found in Titchmarsh's book.
A: The explanation is quite trivial as you can think of the sum of length $t^{1/4}$ as approximately a smoothing of the sum of length $t^{1/2}$ in an interval of length approximately $4 / \log t$. So of course typically the two sums are not too far away. Moreover if the sum of length $t^{1/4}$ is large then there is a nearby large value of the sum of length $t^{1/2}$. 
In particular the phenomenon you're observing has nothing to do with analytic number theory, it's a general fact of harmonic analysis applied to Dirichlet polynomials.
P.S: I don't understand why the other answer keeps getting upvoted while in fact not explaining anything. The equivalent statement of Lindelof stated there is trivial because the "approximating" Dirichlet polynomial is trivially bounded by $t^{\varepsilon}$. So the said equivalence is by the triangle inequality equivalent to $|\zeta(\tfrac 12 + it)| \ll t^{\varepsilon}$ and as such the "equivalence" doesn't involve anything about short Dirichlet polynomials and is not explaining anything... Note that Lindelof does not imply that the polynomial of length t^{1/4} is close to $\zeta(\tfrac 12 +it)$, this statement is simply false point-wise, even though one could conjecture that the ratios of the two Dirichlet polynomials are typically not too far apart. In any case the point-wise approximation can't be because the zeros of the short polynomial are in the first place not on the half-line...  
