AI 中文总结
本文通过对素zeta函数及殆素数生成函数应用相同操作，构造了黎曼R函数的光滑族R_k，证明其展开为Selberg-Delange展开，并利用有限Perron分解确定了精确计数所需的额外围道项。
AI 中文摘要
素zeta函数的实轴跳跃给出了黎曼R函数的Gram级数的积分形式。对经典殆素数生成函数应用相同操作，定义了光滑族R_k，其中R₁=R。我们给出该构造及收敛性论证，并解释为何其在每个固定对数阶的展开为Selberg-Delange展开；该差异源于定义保留了更小的实轴贡献，而非新的渐近系数。有限Perron分解还确定了精确恢复计数所需的额外围道项。
英文摘要
The real-axis jump of the prime-zeta function gives an integral form of Gram's series for Riemann's $R$-function. Applying the same operation to the classical almost-prime generating functions defines a smooth family $R_k$, with $R_1=R$. We give the construction and a convergence argument, and explain why its expansion at every fixed logarithmic order is the Selberg--Delange expansion. The distinction lies in the smaller real-axis contributions retained by the definition, rather than in new asymptotic coefficients. A finite Perron decomposition also identifies the additional contour terms required for exact recovery of the count.
