Let $q \equiv 3 \pmod{4}$ be a prime, $P \geq 1$ and let $N_q(P)$ denote the number of rational primes $p \leq P$ that split in the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$. The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for $N_q(P)$ in the range $q^{1/4+\varepsilon} \leq P \leq q$, for any fixed $\varepsilon>0$. This improves upon what is implied by work of Pollack and Benli-Pollack. The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalently Salie sums). Our estimate has a power saving in the so-called P{o}lya-Vinogradov range, and our methods involve studying an additive energy coming from quadratic residues in $\mathbb{F}_q$. This bilinear form is inspired by the recent \textit{automorphic} motivation: the second moment for twisted $L$-functions attached to Kohnen newforms has recently been computed by the first and fourth authors. So the third part of this paper links the above two directions together and outlines the arithmetic applications of this bilinear form. These include the equidistribution of quadratic roots of primes, products of primes, and relaxations of a conjecture of Erdos-Odlyzko-Sarkozy.

中文翻译：

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用于模平方根和应用的 Weyl 和中的双线性形式

令 $q \equiv 3 \pmod{4}$ 为素数，$P \geq 1$ 并令 $N_q(P)$ 表示在虚二次域 $ 中分裂的有理素数 $p \leq P$ \mathbb{Q}(\sqrt{-q})$。本文的第一部分在 $q^{1/4+\varepsilon} \leq P \leq q$ 范围内为 $N_q(P)$ 建立了各种无条件和有条件（在 Siegel 零存在下）下界，对于任何固定的 $\varepsilon>0$。这改进了 Pollack 和 Benli-Pollack 的工作所暗示的内容。本文的第二部分致力于证明涉及模平方根的 Weyl 和（相当于 Salie 和）的双线性形式的估计。我们的估计在所谓的 P{o}lya-Vinogradov 范围内具有节能效果，我们的方法涉及研究来自 $\mathbb{F}_q$ 中二次残差的附加能量。这种双线性形式的灵感来自最近的 \textit{automorphic} 动机：附加到 Kohnen newforms 的扭曲 $L$ 函数的第二矩最近由第一和第四作者计算。所以本文的第三部分将上述两个方向联系在一起，并概述了这种双线性形式的算术应用。其中包括素数的二次根的等分分布、素数的乘积以及 Erdos-Odlyzko-Sarkozy 猜想的松弛。