Distribution of square roots mod 1 I was wondering about the distribution of $\sqrt{p}$ mod $1$ this morning, as one does while brushing one's teeth.  I remembered the paper of Elkies and McMullen (Duke Math. J. 123 (2004), no. 1, 95–139.) about $\sqrt{n}$ mod $1$, but hadn't really thought about it before.  
Question 1:  is $\sqrt{p}$ equidistributed mod $1$, as $p$ varies over all prime numbers?  Is this known?  Within range of current techniques?
Question 2:  What about subtler statistics of $\sqrt{p}$ (and $\sqrt{n}$) mod $1$?  
I made three plots, giving histograms of $\sqrt{n}$ mod $1$ (for natural numbers up to 100,000) and $\sqrt{p}$ mod $1$ (for primes up to 1 million) and (for comparison) a histogram of 100,000 samples drawn uniformly at random from {0,1,...,999}.  Here they are for your enjoyment.



There's some wild stuff going on, I think!  
Question 2(a):  What's up with these sharp peak/valleys at rational numbers, in the distribution of $\sqrt{n}$ mod $1$?  They are especially prominent at fractions of the form $a / 2^{e}$.  How tall are these peaks near rational numbers?  They persist when sampling from the primes, i.e., in the distribution of $\sqrt{p}$ mod $1$ too.
Question 2(b):  Outside of those funky spots in 2(a), the distribution of $\sqrt{n}$ mod $1$ is far flatter than one would expect, e.g., from samples drawn uniformly at random as displayed in the bottom histogram.  This must have been noticed and quantified before... what's the relevant quantitative result here?
Question 2(c):  The distribution of $\sqrt{p}$ mod 1 displays the same funky spots near rational numbers, but otherwise seems much closer (in noise-volume) to the random samples at the bottom.  Maybe for a larger sample, the funkiness goes away... I don't know.  Explanations or conjectures are welcome.
Question 3:  These seem like natural images to look at.  If you know a reference where others have drawn such pictures or studied similar phenomena, I'd love to take a look!  
-------------Update after answers below-----------------
It looks like the answer to Question 1 is YES.  Lucia's answer below explains this, and also some of the flatness evident in the $\sqrt{n}$ distribution mod 1.
Igor and Aaron discuss the "spikes" around rational numbers.  This seems related to binning:  if our bins have width 1/1000, we see spikes at multiples of 1/2, 1/4, 1/5, 1/10, etc., related to divisors of 1000.  Here's a new picture, which might help us understand the behavior of the distribution of $\sqrt{n}$ mod 1 near rational numbers.  I've intentionally drawn the bins so that their endpoints lie on rational numbers with denominator up to 60.  (I call this Farey-binning).  This seems to bring the "spikes" around rational numbers down to the same size (independent of denominator). 
I think I'll accept Lucia's answer soon, because it answers the most direct Question 1.  But more insights are welcome.

 A: For the spikes, note that in your picture, the predicted number of points in the bin around 0 is 100, but there are 300 squares in your range. Similarly, there are 75 quarter-squares (where by quarter squares I mean numbers of the form $(n^2-1)/4,$ etc, so I think it explains some of the rational spiking. 
A: It is known that the distribution of the pair correlations of $\sqrt{n}$ mod 1 is asympotitically Poissonian (that is, "random"). See [1]. Note that having Poissonian pair correlations implies being equidistributed [2], so this is a stronger result than the equidistribution result.
I don't know if a similar result is known for $\sqrt{p}$. Also, I don't know about results for higher correlations or neighbor spacings.
You might also want to check the related paper [3].
[1] El-Baz, Daniel; Marklof, Jens; Vinogradov, Ilya. The two-point correlation function of the fractional parts of \sqrt{n} is Poisson. Proc. Amer. Math. Soc. 143 (2015), no. 7, 2815–2828. 
[2] https://arxiv.org/abs/1612.05495
[3] https://www.maths.bris.ac.uk/~majm/bib/nato.pdf
A: You can certainly use Vinogradov's method to show that $\sqrt{p}$ is equidistributed $\pmod 1$.  I haven't thought about more subtle properties, such as the gap spacing considered by Elkies and McMullen (or your other questions).   
For the equidistribution, by Weyl's criterion it is enough to show cancellation in sums of the form 
$$ 
\sum_{n\le x} \Lambda(n) e(k\sqrt{n}) 
$$ 
for non-zero integers $k$.   This is exactly the kind of sum to which Vinogradov's method applies.  For example, see Exercise 2 on page 348 of Iwaniec-Kowalski which invites you to show that this sum is $\ll_k x^{\frac 56+\epsilon}$.  Sums like this also appeared in the IHES paper of Iwaniec, Luo and Sarnak, where they show that better bounds for this sum (like $O(x^{\frac 12+\epsilon})$) have implications for the Riemann hypothesis for $GL(2)$ $L$-functions. 
One should expect the exponential sum over primes above to be on the scale of $O(x^{\frac 12+\epsilon})$.  This is in keeping with the plots for $\sqrt{p}$ looking like random noise.  To see why $\sqrt{n}$ looks different and more flat, note that the number of $n\le N^2$ with $\{ \sqrt{n} \} \in (\alpha,\beta)$ is given by 
$$ 
\sum_{k\le N} \sum_{(k+\alpha)^2 < n <(k+\beta)^2} 1 = \sum_{k\le N} (\lfloor 2k\beta+\beta^2 \rfloor - \lfloor 2k \alpha + \alpha^2\rfloor). 
$$ 
Since the distribution of $\{ 2k\alpha+\alpha^2\}$ (and similarly for $\beta$) is extremely regular, one should expect this to be nailed down much more precisely than for primes.  
Finally, suppose for example that $\alpha=a/q$ is a rational number (in lowest terms) with small denominator $q$, which let us assume odd for simplicity.  Write $\alpha^2 = b/q + c/q^2$ with $0<c <q$.  Note that $\{ 2k\alpha+\alpha^2\}$ will run over $c/q^2$, $1/q+c/q^2$, $\ldots$, $(q-1)/q+c/q^2$, and its average value will be $(q-1)/(2q) + c/q^2$.  This can be noticeably different from the average value of $\{ x\}$, which is $1/2$, explaining the "spikes" near small rational numbers. 
A: Here are some quick details about the imbalance at $\frac12,$ It shouldn't be hard to generalize. Suppose specifically that there are $1000$ equal size bins on $[0,1].$ On pass $t$  we put the fractional parts of the $2t$ irrational square roots $\lfloor\sqrt{t^2+k}\rfloor$ for $1 \le k \le 2t$ into the appropriate bins.
On each pass most bins get as many new things as on the previous pass. Exactly two bins get one more thing than before. On pass $t$ the following can be shown to happen for the bins $A=[0.499,0.5]$ and $B=[0.5,0.501]:$ For $0 \leq t \leq 124$ neither one gets anything. Let $t=125+500q+r$


*

*For $0 \leq r \leq 249$ $A$ gets $q+1$ new things but $B$ only gets $q.$

*For $250 \leq r \leq 499$ both $A$ and $B$ get $q+1$ new things.


Let $|A|$ and $|B|$ denote the counts in the bins after a certain pass. Then it follows that 


*

*For $0 \leq r \leq 249,$ $|A|-|B|=250(q+1)$ and $\frac{|A|}{|B|}=\frac{q}{q-1}.$ 

*For  $250 \leq r \leq 499,$ $|A|-|B|$ is increasing and $\frac{|A|}{|B|}$ is decreasing.
