Refine counts and speeds of convergence of zeros; classify exponential-rate zeros; extend to other groups and even levels

Improve or determine exactly, for Γ(N) with odd N, the cardinalities of the sets of zeros with prescribed convergence speeds to the unit-arc described in Theorem 8, fully identify all zeros that converge at exponential speed, and extend the precise convergence-speed analysis to non-principal congruence subgroups and to even levels N.

Background

For odd N, the paper gives lower bounds and asymptotics for the number of zeros approaching the unit arc at rate ≍1/k, identifies rare families converging exponentially fast to special points (i and ρ), and proves angular equidistribution with discrepancy O((log N)/k).

The authors ask for sharper, possibly exact, counts of zeros at different rates, a complete classification of those with exponential convergence, and analogous results beyond principal groups and beyond odd N.