The quartic Gauss-sum $X^{3/4}$ bias conjecture over Gaussian primes

Let $e(x):=e^{2\pi i x}$ for $x\in\mathbb{R}$ and $\check{e}(z):=e(z+\overline{z})$ for $z\in\mathbb{C}$. Let $\mathbb{Q}(i)$ be the Gaussian quadratic field with ring of integers $\mathbb{Z}[i]$, let $\lambda:=1+i$, and for $c,\nu\in\mathbb{Z}[i]$ with $(c,\lambda)=1$ define

$$\widetilde{g}_4(\nu,c):=\frac{1}{\sqrt{N(c)}}\sum_{d\pmod{c}}\left(\frac{d}{c}\right)_4\check{e}\left(\frac{\nu d}{c}\right),$$

where $\left(\frac{\cdot}{c}\right)_4$ is the quartic symbol, $\widetilde{g}_4(c):=\widetilde{g}_4(1,c)$, and $\Lambda(c)$ denotes the von Mangoldt function on $\mathbb{Z}[i]$. The quartic Gauss-sum $X^{3/4}$ bias conjecture. For $\beta\in\{1,1+\lambda^3\}\pmod{4}$, there exists a constant $b_\beta\neq 0$ such that for any $\varepsilon>0$ and $\ell\in\mathbb{Z}$,

$$\sum_{\substack{c\in\mathbb{Z}[i]\N(c)\leq X\c\equiv\beta\pmod{4}}}\widetilde{g}_4(c)\left(\frac{\overline{c}}{|c|}\right)^\ell\Lambda(c)= \begin{cases} b_\beta X^{3/4}+O_\varepsilon(X^{1/2+\varepsilon})&\text{if }\ell=0,\\ O_{\varepsilon,\ell}(X^{1/2+\varepsilon})&\text{if }\ell\neq0, \end{cases}$$

as $X\to\infty$. This conjecture predicts a main-term bias of order $X^{3/4}$ in the first moment of quartic Gauss sums over primes, while all nontrivial angular moments are smaller. The paper proves a substantially weaker upper bound of order $X^{5/6+\varepsilon}+X^{3/4+\varepsilon}|\ell|^{3/2+\varepsilon}$; the stated asymptotic remains unproved.