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Is it known that the primes on the Ulam prime spiral distribute themselves equally in sectors around the origin? To be specific, say the quadrants? (Each quadrant is closed on one axis and open on the other.) For example, in the $50 \times 50$ spiral below, I count the number of primes in the four quadrants to be $(103,96,88,86)$ ($\sum=373$), leading to ratios $(0.276,0.257,0.236,0.231)$:
           Prime Spiral
Continuing to $1500 \times 1500$, which includes $166169$ primes, I count $(41561,41528,41685,41395)$ leading to $(0.2501,0.2499,0.2509,0.2491)$:
           Plot
I earlier asked this question at MSE, where user Fred Daniel Kline reported carrying out the computation to include 1.7 billion primes, when the quadrant ratios fall within $\frac{1}{4} \pm 10^{-6}$.

Empirically there is a convergence to $\frac{1}{4}$, but I wonder if this has been proven? Thanks!

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My numbers involved different quadrant boundaries (each different by one). mathematica.stackexchange.com/questions/6219/… –  Fred Kline Jul 13 '12 at 0:54
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2 Answers 2

up vote 16 down vote accepted

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).)

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That is an incisive observation that "the fractional part of $p^{1/2}$ asymptotically determines" the location along the spiral. Thanks for resolving this! –  Joseph O'Rourke Jul 13 '12 at 10:15
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You stated that distribution of primes within <a href="mathematica.stackexchange.com/q/8113/973">; $\left[n, n+n^{\frac{1}{2}}\right]$</a> is known by the PNT. If that could be adapted to my algorithm at the link, it would imply Oppermann's Conjecture. Could you point me to the appropriate literature? –  Fred Kline Jul 14 '12 at 1:03
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This is only currently known on the average (i.e. for almost all n, rather than for all n). See for instance the Iwaniec-Kowalski book (in the chapter on zero-density estimates). –  Terry Tao Jul 14 '12 at 5:33
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There's only 7 prime quadrant numbers under the number 2000

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I don't know what this means. What's a prime quadrant number, and what does it have to do with Joseph's question? –  Gerry Myerson Feb 26 '13 at 11:09
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