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Determine optimal Kluyver-sum constants for all congruence-group types

Determine, for every finite subgroup G of SL_2(Z/NZ) (equivalently, for every congruence subgroup Γ of level N), the optimal Kluyver-sum constant κ_Γ (or κ_G) defined as the maximum over row vectors v,w in (Z/NZ)^2 of the least positive j with ρ_G^{v,w}(j) ≠ 0, and in particular resolve the case of prime level N where κ_Γ is conjecturally either 1 or N by classifying it for each subgroup type up to conjugacy.

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Background

The paper introduces Kluyver sums ρ_G{v,w}(j) attached to subgroups G ≤ SL_2(Z/NZ) and defines κ_G as the minimal positive index j of non-vanishing, maximized over v,w. These constants govern the zero-free region for Eisenstein series and their translates, and are computed explicitly for several standard families (e.g., principal congruence subgroups and some related groups), with upper bounds in general.

The authors emphasize that, beyond the cases treated (e.g., Γ(N) with parity conditions on N, Γ1(N), Γ_0(N), and the theta group), finding the exact κΓ for general subgroup types—especially at prime level, where subgroups are well classified—is still unresolved.

References

We list some open problems. Study Kluyver sums and find the optimal constants \kappa_\Gamma for congruence groups of any type in \mathrm{SL}_2(\mathbb Z/N\mathbb Z), e.g., for N prime, when they are either 1 or N; notice that, by Lang's Theorem, conjugacy classes of such subgroups for N prime are the same over \mathbf F_p and \overline{\mathbf F}_p, and all conjugacy classes of subgroups and representatives for each class are know, first in the work of Gierster [Gierster]. See Example \ref{exotic} for one case.

— Geodesic clustering of zeros of Eisenstein series for congruence groups (2509.16108 - Santana et al., 19 Sep 2025) in Section: Open problems (final section)