Discoveries / number-theory

Erdős Problem #390: the second-order constant for $f(n)-2n$

The headline is the constant, and its two halves have different histories. The lower bound, $\liminf (f(n)-2n)/(n/\log n) \ge 4029639598/25970038185$, is not new here: it is Mausberg's thirteen-layer valuation cut, posted to the erdosproblems.com forum in May 2026 and credited as such in the paper. Its author wrote there that it "does not prove an upper bound, nor does it prove that an asymptotic constant exists." The novelty is the matching upper bound, so the claim is that the thirteen-layer bound is exactly tight. It is assembled from an exact cofactor-allocation certificate, central-binomial anchors, a guarded rough-signature selector, a friable-number covariance bridge, a finite-band tangent correction, and column-sparse rounding. That construction is what a reader should scrutinize; everything else is inherited or machine-checked. erdosproblems.com still lists #390 as open, and the paper calls itself a proposed solution.