Linowitz–Thompson conjecture for the average first negative Fourier coefficient

Let $D_1,D_2$ range over fundamental discriminants with $|D_1D_2|\to obreak\ter x$, and let $\eta(D_1,D_2)$ be the smallest \prime $p$ such that the sign of the corresponding Fourier coefficient satisfies $\operatorname{sgn}(\sigma_{\chi_1,\chi_2}^{k-1}(p))=-1$. Define

$$\theta=\sum_{k=1}^{\infty}\frac{p_k^2(p_k+2)}{2(p_k+1)^2}\prod_{j=1}^{k-1}\frac{2+p_j(p_j+2)}{2(p_j+1)^2}\approx 3.9750223902\ldots,$$

where $p_k$ is the $k$-th prime. Linowitz–Thompson conjecture. The average of $\eta(D_1,D_2)$ satisfies

$$\frac{\sum_{|D_1D_2|\leqslant x}\eta(D_1,D_2)}{\sum_{|D_1D_2|\leqslant x}1}\xrightarrow[x\to\infty]{}\theta.$$

The conjecture was motivated by the distribution of signs of Fourier coefficients of Eisenstein series newforms. The paper explains that the heuristic is not uniform in the relevant prime index and proves that the limiting average differs from this conjectured constant by a small nonzero contribution.