The negative bias conjecture for Gaussian-prime representations by sums of squares

Let

$$D_1(x)=\lvert\{p<x:p=a^2+4b^2,\ \lvert a\rvert>\lvert2b\rvert\}\rvert-\lvert\{p<x:p=a^2+4b^2,\ \lvert a\rvert<\lvert2b\rvert\}\rvert.$$

Here $p$ ranges over primes, and logarithmic scale means the measure induced by $\mathrm{d}x/x$; equivalently, the logarithmic density of a set $\mathcal P\subset[1,\infty)$ is defined using

$$\overline{\delta}(\mathcal P)=\limsup_{Y\to\infty}\frac1Y\int_0^Y\mathbf 1_{\mathcal P}(e^y)\,\mathrm{d}y, \qquad \underline{\delta}(\mathcal P)=\liminf_{Y\to\infty}\frac1Y\int_0^Y\mathbf 1_{\mathcal P}(e^y)\,\mathrm{d}y.$$

Negative bias conjecture. There is a bias towards negative values in the distribution of $D_1$: for more than half of the $x\in[2,\infty)$ in logarithmic scale, more than half of the primes below $x$ can be written as a sum of two squares with the even square larger than the odd square. Moreover,

$$D_1(x)=\Omega_{\pm}\left(\frac{\sqrt{x}}{\log x}\right),$$

so $D_1$ changes signs infinitely often. This conjecture describes a discrepancy in the expected equidistribution of the two possible size orderings in representations $p=a^2+4b^2$. Under the Generalized Riemann Hypothesis, Hecke's results give the bound $D_1(x)=O_{\epsilon}(x^{1/2+\epsilon})$, while the asserted logarithmic bias and the stated oscillation are part of the conjectural picture.