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Erdős #969 · number theory

Let Q(x)Q(x) count the number of squarefree integers in [1,x][1,x]. Determine the order of magnitude in the error term in the asymptotic

Q(x)=6π2x+E(x).Q(x)=\frac{6}{\pi^2}x+E(x).
MSC 11 · Number theory

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Problem statement

Let Q(x)Q(x) count the number of squarefree integers in [1,x][1,x]. Determine the order of magnitude in the error term in the asymptotic

Q(x)=6π2x+E(x).Q(x)=\frac{6}{\pi^2}x+E(x).

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Background

It is elementary to prove E(x)≪x1/2E(x)\ll x^{1/2}, and the prime number theorem implies o(x1/2)o(x^{1/2}). The best known unconditional upper bound is of the shape x1/2−o(1)x^{1/2-o(1)}, due to Walfisz [Wa63]. Evelyn and Linfoot [EvLi31] proved that

E(x)≫x1/4,E(x) \gg x^{1/4},

and this is likely the true order of magnitude. The Riemann Hypothesis would follow from E(x)≪x1/4E(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)≪x1135+o(1),E(x)\ll x^{\frac{11}{35}+o(1)},

due to Liu [Li16].

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The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #969 link

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