Problem Statement
Let $Q(x)$ count the number of squarefree integers in $[1,x]$. Determine the order of magnitude in the error term in the asymptotic\[Q(x)=\frac{6}{\pi^2}x+E(x).\]
Categories:
Number Theory
Progress
It is elementary to prove $E(x)\ll x^{1/2}$, and the prime number theorem implies $o(x^{1/2})$. The best known unconditional upper bound is of the shape $x^{1/2-o(1)}$, due to Walfisz [Wa63]. Evelyn and Linfoot [EvLi31] proved that\[E(x) \gg x^{1/4},\]and this is likely the true order of magnitude. The Riemann Hypothesis would follow from $E(x)\ll x^{1/4}$.The true order of magnitude is unknown even assuming the Riemann Hypothesis. Conditional on this assumption, the best known upper bound is\[E(x)\ll x^{\frac{11}{35}+o(1)},\]due to Liu [Li16].
Source: erdosproblems.com/969 | Last verified: January 19, 2026