Montgomery's integer Chebyshev conjecture

For a compact set $K\subset\mathbb{R}$ , define the integer Chebyshev constant by $t_{\mathbb{Z}}(K)=\lim_{n\to\infty}\inf\{\lVert P\rVert_K^{1/n}:0\ne P\in\mathbb{Z}[x],\ \deg P\le n\}$ , where $\lVert P\rVert_K=\max_{x\in K}|P(x)|$ . Montgomery's conjecture asks for the exact value of $t_{\mathbb{Z}}([0,1])$ and asserts that it equals a specific exact constant. The supplied sources do not state the conjectured identity itself; they record only the bounds $0.4213<t_{\mathbb{Z}}([0,1])<0.42291334$ and that the 2026 preprint claims to prove Montgomery's conjecture.

References

Primary source

arXiv

Additional references

Limit distribution of algebraic integral points on curves — arXiv — Binggang Qu, Chengyuan Yang

Posted 6 days ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed solved

A September 2026 preprint claims to settle the conjecture, but its result has not been independently checked.

Montgomery conjectured an exact identity for the integer Chebyshev constant of $[0,1]$ . Earlier surveys described this identity as open and gave only numerical bounds.

Known results

The integer Chebyshev constant satisfies $0.42072638\ldots < t_{\mathbb Z}([0,1]) \le 0.42347945$ .
Later work narrowed this to $0.4213 < t_{\mathbb Z}([0,1]) < 0.42291334$ .
The exact value was not known for any segment of length less than $4$ , including $[0,1]$ .
Integer Chebyshev polynomials have endpoint factors with multiplicity exceeding $0.26n$ for sufficiently large $n$ .

September 2026 claimed solution

Binggang Qu and Chengyuan Yang’s preprint Limit distribution of algebraic integral points on curves develops limit distributions and applies them to the integer Chebyshev constant of $[0,1]$ , claiming Montgomery’s identity. The preprint is unrefereed, and no independent verification or mathematical assessment of the claimed resolution was found.

Current status (as of October 2026): Montgomery’s conjecture has a claimed solution in an unrefereed preprint, but the identity remains unverified and the problem is not settled.

Sources

No solutions have been posted yet.