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Statement

Conjecture 3 (p. 167): "For every $\varepsilon>0$ there is a constant $K=K(\varepsilon)$ such that $D(N)\le KN(\ln N)^{1+\varepsilon}$." Here $D(N)=\max_{0<a<N}D(a,N)$ is the least possible largest denominator as defined on the Theorem 1 page.

This is the question of Problem 305, which the site writes as $D(b)\ll b(\log b)^{1+o(1)}$; the 1980 monograph (p. 38) restates it as "for every $\varepsilon>0$, $D(b)\le c(\varepsilon)b(\log b)^{1+\varepsilon}$". Yokota's paper On a problem of Bleicher and Erdős, J. Number Theory 30 (1988), 198--207, is the site's solving citation; Liu and Sawhney's Theorem 1.5 gives the current bound $b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}$.

Source. Bleicher--Erdős, J. Number Theory 8 (1976), Conjecture 3 on printed p. 167 (PDF p. 11), among four conjectures closing the paper (the others concern the constant in Lemma 2, a submultiplicativity property of $D$, and lacunary denominator sequences). Read on the page image.

Read depth. Claims checked: the statement was read clause by clause on the page image. It is a conjecture; there is no proof to check in this paper.

Dependencies

None.

Bears on

• Problem 305: the origin of the problem's question.