Fixed-normalized-window theorem at the endpoint α=1

Determine whether the fixed-normalized-window theorem for the prime-modulus Dirichlet family holds at the endpoint α=1, corresponding to the shrinking interval I=((log q)^{-1},2(log q)^{-1}], including the stated lower bounds for simple critical-line zeros, distinct critical-line zero points, and all distinct nontrivial zero points.

Background

The paper proves its zero-proportion results only for fixed 0<α<1, with interval length T=(log q)^{-α}. At α=1, the total family zero scale falls to order q, so several error terms that are negligible for α<1 become comparable to the main term. In addition, the sampled Gabor dimension no longer tends to infinity, and the near-line exterior-tail estimate does not decay by increasing a fixed Schwartz order.

Consequently, the methods developed in the paper do not establish the analogous theorem at α=1, but the authors also do not disprove it. The unresolved issue is whether the corresponding fixed-normalized-window result nevertheless holds by this or another method.