AI 中文总结
研究为无平方因子级全纯新形式建立联合水平 - 权重波动定律，从精确迹公式出发，经权重求和、截断\(L\)值、平均周期函数及诺伊曼求和等步骤，无需\(L\)函数零点假设，得到收敛到特定测度的结果。
AI 中文摘要
我们为无平方因子级的全纯新形式建立了一个无条件的联合水平 - 权重波动定律。水平\(N\)平均大小为\(X\)，偶数权重\(k\)平均大小为\(K\)，素数在坐标\(p\)除以\(N\)乘以\(((k - 1)/(4\pi))^2\)中度量。归一化后，根数加权素数迹收敛到一个支撑在有理平方上的局部有限测度，具有明确给出的欧拉乘积质量。对于\(X^{\epsilon} \leq K \leq X^{1 - \epsilon}\)收敛是一致的，因此对于每个固定的\(0 < \rho < 1\)，\(K = X^{\rho}\)时也成立。从精确迹公式出发，我们首先对权重求和，用伯吉斯界截断所得的二次\(L\)值，并在无平方因子水平上对剩余的周期函数求平均。诺伊曼求和将所得的贝塞尔级数转化为原子测度。未使用关于狄利克雷或自守\(L\)函数零点的假设。
英文摘要
Let (N) range over squarefree levels (N \asymp X), and let the even weight (k) vary in a smooth window (k \asymp K). We study natural root-number-weighted traces of Hecke eigenvalues at primes (p \asymp XK^2). First, uniformly for (X^{\varepsilon_0} \leq K \leq X^{3-\varepsilon_0}), we prove a fixed-prime asymptotic whose main term is expressed through Zubrilina's murmuration density. The exponent (3) is the endpoint of our absolute treatment of the nonzero Poisson frequencies. Second, after averaging the primes with logarithmic weight, we obtain unconditionally, throughout every fixed polynomial range (X^{\varepsilon_0} \leq K \leq X^{A_0}), an explicit atomic limiting measure. The fixed-prime argument groups the local Fourier expansion by exact additive conductor and uses Burgess truncation, whereas the prime-averaged argument applies a smooth Barban-Davenport-Halberstam estimate.
Comments46 pages, 1 figure
