发表机构
Clemson University（克莱姆森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对标准和对称平方L函数的素二次扭曲，研究其一致非零性，证明满足特定条件的素数比例趋于零，还得到相关基的生成与密度一结果，核心方法涉及随机欧拉乘积的非集中定理。
AI 中文摘要
设$S_k$为${\rm SL}_2(\mathbb Z)$的权为$k$的尖形式空间，其中$\dim S_k\geq1$，并设$\chi_p=\left(\frac{\cdot}{p}\right)$为奇素数$p$。对于满足${\rm Re}(s)>(k+1)/2$的固定$s$，我们证明满足$L(h,\chi_p,s)=0$的素数$p\leq X$的比例随$X\to\infty$趋于零，且在非零$h\in S_k$上一致成立。我们还证明了当${\rm Re}(s)>k$时，归一化赫克本征形式的扭曲对称平方$L$值的非平凡线性组合的类似结论。证明的主要要素是成对局部分离条件下随机欧拉乘积的非集中定理。作为应用，我们在$S_k$中得到与扭曲标准和对称平方$L$函数相关的核的生成基和密度一结果，当$\dim S_k=2$或$3$时，在指定右半平面中有显式基准则。我们证明几乎每个$\dim S_k$素数的元组都从每个非中心临界指数处的扭曲周期给出$S_k^{\ast}$的基。我们还证明了在素数水平处从艾森斯坦级数的兰金-科恩括号的迹得到的$S_k$的有理基的密度一结果。
英文摘要
Let $S_k$ be the space of cusp forms of weight $k$ for ${\rm SL}_2(\mathbb Z)$, with $\dim S_k\geq1$, and let $χ_p=\left(\frac{\cdot}{p}\right)$ for odd primes $p$. For fixed $s$ with ${\rm Re}(s)>(k+1)/2$, we prove that the proportion of primes $p\leq X$ satisfying $L(h,χ_p,s)=0$ tends to zero as $X\to\infty$, uniformly over nonzero $h\in S_k$. We also prove an analogous statement for nontrivial linear combinations of twisted symmetric square $L$-values of the normalized Hecke eigenforms when ${\rm Re}(s)>k$. The main ingredient of the proof is a non-concentration theorem for random Euler products under a pairwise local separation condition. As applications, we obtain spanning and density-one basis results in $S_k$ for the kernels associated with twisted standard and symmetric square $L$-functions, with explicit basis criteria in specified right half-planes when $\dim S_k=2$ or $3$. We show that almost every tuple of $\dim S_k$ primes gives a basis of $S_k^{\ast}$ from twisted periods at every noncentral critical index. We also prove a density-one result for rational bases of $S_k$ obtained from traces of Rankin-Cohen brackets of Eisenstein series at prime levels.
