The half-integral-weight Linnik–Selberg conjecture

Let $k=\frac12$ or $k=-\frac12$, and let $u$ be a weight $k$ multiplier system on $\Gamma_0(N)$. Let $S(m,n,c,\nu)$ be the corresponding half-integral-weight Kloosterman \sum, let $\widetilde m$ and $\widetilde n$ denote the associated parameters, and let $\tau_j(m,n)$ be the coefficients attached to normalized eigenforms of the weight $k$ hyperbolic Laplacian $\Delta_k$. Assume that $\Delta_k$ has no eigenvalue in $\left(\frac3{16},\frac14\right)$. The half-integral-weight Linnik–Selberg conjecture. Under these assumptions,

$$\sum_{N\mid c\leq X}\frac{S(m,n,c,\nu)}{c}-2X^{\frac12}\sum_{r_j=\frac i4}\tau_j(m,n)\ll_{N,\nu,\varepsilon}|\widetilde m\widetilde nX|^\varepsilon,$$

where the second sum runs over normalized eigenforms of $\Delta_k$ with eigenvalue $\frac14+r_j^2=\frac3{16}$. This is the half-integral-weight analogue of the Linnik–Selberg spectral-gap formulation; the supplied text does not establish whether the conjectural spectral condition is known in this setting.