Odoni [Od81] proved this is false, and that\[A(x) \gg \exp\left(c\frac{\log\log\log x}{\log\log x}\right)\frac{x}{\sqrt{\log x}}\]for some constant $c>0$. Estimates for $A(x)$ have been improved by Baker and Brudern [BaBr94], Blomer [Bl04], and most recently by Blomer and Granville [BlGr06], who proved\[A(x)= (\log\log x)^{O(1)}\frac{x}{(\log x)^{\alpha}}\]where $\alpha=1-2^{-1/3}\approx 0.206299$.

See also [940].

View the LaTeX source

This page was last edited 15 October 2025.