Prime spiral distribution into quadrants

I haven't looked at the papers in detail, but apparently this paper of Wolke and also this paper of Stux establish that the square roots $p^{1/2}$ of primes are uniformly distributed modulo 1. As the fractional part of $p^{1/2}$ asymptotically determines the argument along the Ulam spiral, this should give the desired uniform distribution. (Such results also be obtainable from the known versions of the prime number theorem in short intervals such as $[x,x+x^{1/2}]$ on the average, which in turn should follow from the known zero density theorems for the zeta function (as discussed, for instance, in the Iwaniec-Kowalski book).)