Classification of irreducible balanced coprime pairs

Determine the complete classification of irreducible balanced coprime pairs of positive-integer tuples satisfying Landau’s criterion, including pairs of arbitrary height.

Background

The paper discusses classifications of special classes of pairs satisfying Landau’s criterion, including Bober’s complete classification in height one and Soundararajan’s partial classification in height two. A complete classification for arbitrary height would clarify the range of q-factorial ratios to which the q-rious positivity and unimodality conjectures apply.

The authors note that an exhaustive case-by-case approach is unlikely to settle the positivity conjecture because the general classification itself is unresolved and may be intractable for arbitrary height.

References

Of course, since the classification of irreducible balanced coprime pairs satisfying (1) is still completely open (and for arbitrary height is almost surely intractable), a case-by-case approach will never settle the conjecture.

— $q$-rious unimodality  (2502.03993 - Warnaar et al., 6 Feb 2025) in Section preceding Conjecture 5, Introduction