Effective non-vanishing conjecture for the first two moments without twist

Let $T$ and $\varPi$ be parameters, and let $\varepsilon>0$. Denote by $\mathop{\mathrm{C}}_1(1)$ and $\mathop{\mathrm{C}}_2(1,1)$ the first and second moments considered in the paper, and let $\gamma_1$ and $\gamma_0'$ be the constants occurring in their main terms. For any

$$T^{\varepsilon}\leqslant \varPi\leqslant T^{1-\varepsilon},$$

Effective non-vanishing conjecture. One has

$$\mathop{\mathrm{C}}_1(1)=\frac{2}{\pi\sqrt{\pi}}\varPi T+o(\varPi T),\qquad \mathop{\mathrm{C}}_2(1,1)=\frac{4}{\pi\sqrt{\pi}}\varPi T\bigl(\gamma_1\log T+\gamma_0'\bigr)+o(\varPi T).$$

The conjecture extends the ranges in which the paper proves asymptotic formulas for these untwisted moments, asserting that their predicted main terms remain valid throughout the full range $T^{\varepsilon}\leqslant \varPi\leqslant T^{1-\varepsilon}$.