2
$\begingroup$

Let $\Psi (x,y)$ denote the number of $y$-smooth integers less than or equal to $x$.

I know the precise results about $\Psi (x,y)$ in the literature (with good error terms), but what are some helpful "order of magnitudes" to keep in mind, that give a representation of what's going on when $y$ varies?

$\endgroup$

1 Answer 1

6
$\begingroup$

Using the estimation

$$\Psi(x,y) = x \rho(u) \Big( 1 + O\Big(\frac{\log(u+1)}{\log y}\Big)\Big) \qquad (x \geq 3,\ e^{(\log_2x)^{5/3+\varepsilon}} \leq y \leq x)$$ (see §III.5.5 of [T15])

or

$$\Psi(x,y) \ll x e^{-u\log u} + \sqrt{x} \qquad (x \geq2,\ y \geq 2)$$ (see §III.5.6 of [T15]), we can get the following order of magnitudes for the number of $y$-smooth numbers up to $x$:

$$ \begin{array}{c|c|c} y & \log y & u = \frac{\log x}{\log y} & \Psi(x,y) \\ \hline x & \log x & 1 & x\\ x^\alpha & \alpha \log x & \frac1\alpha & \sim \rho(1/\alpha) x \\ e^{\frac{(1-\varepsilon) (\log x) (\log_3x)}{\log_2 x}} & \frac{(1-\varepsilon)(\log x) (\log_3x)}{\log_2 x} & \frac{\log_2 x}{(1-\varepsilon)\log_3 x} & \ll \frac{x}{\log x} \\ e^{\frac{\log x}{\log_2 x}} & \frac{\log x }{\log_2 x} & \log_2 x & \ll \frac{x}{(\log x)^{\log_3 x}} \\ e^{\sqrt{\log x}} & \sqrt{\log x} & \sqrt{\log x} & \ll \frac{x}{(\log x)^{(1/2)\sqrt{\log x}}} \\ e^{(\log_2 x)^{5/3}} & (\log_2 x)^{5/3} & \frac{\log x}{(\log_2 x)^{5/3}} & \ll x^{1 - \frac{1-\varepsilon}{(\log_2 x)^{2/3}}}\\ (\log x)^{10} & 10\log_2 x & \frac{\log x}{10 \log_2 x} & \ll x^{0.9+\varepsilon} \\ (\log x)^2 & 2\log_2 x & \frac{\log x}{2 \log_2 x} & \ll x^{0.5+\varepsilon} \\ \log x & \log_2 x & \frac{\log x}{\log_2 x} & \ll \sqrt{x} \\ \end{array} $$

[T15] G. Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd edition, AMS, 2015.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.