论文标题
流畅的整数和Dickman $ρ$功能
Smooth integers and the Dickman $ρ$ function
论文作者
论文摘要
我们为$ψ（x，y）$建立了一个渐近公式，其形状为$ xρ（\ log x/\ log y）$ times校正因子。这些因素考虑了Zeta Zeros和Prime Powers的贡献，该公式可以被视为$ψ（x，y）$的（近似）明确公式。使用此公式，我们证明了$ψ（x，y）$的振荡结果，该结果解决了$ψ（x，y）\ asympxρ（\ log x/\ log y）$的有效性范围内的希尔德布兰德问题。我们还解决了关于$ψ（x，y）\ gexρ（\ log x/\ log y）$的有效性范围范围的问题。 在途中，我们改善了$ψ（x，y）$的经典估计，并在Riemann假设上，发现了$ y =（\ log x）^{3/2+o（1）} $的$ψ（x，y）$的意外阶段过渡。
We establish an asymptotic formula for $Ψ(x,y)$ whose shape is $x ρ(\log x/\log y)$ times correction factors. These factors take into account the contributions of zeta zeros and prime powers and the formula can be regarded as an (approximate) explicit formula for $Ψ(x,y)$. With this formula at hand we prove oscillation results for $Ψ(x,y)$, which resolve a question of Hildebrand on the range of validity of $Ψ(x,y) \asymp xρ(\log x/\log y)$. We also address a question of Pomerance on the range of validity of $Ψ(x,y) \ge x ρ(\log x/\log y)$. Along the way we improve classical estimates for $Ψ(x,y)$ and, on the Riemann Hypothesis, uncover an unexpected phase transition of $Ψ(x,y)$ at $y=(\log x)^{3/2+o(1)}$.
