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Determine Hausdorff-limit zeros for Hecke Eisenstein series G_k^{A,B}

Determine the Hausdorff limit of the zero sets in the standard fundamental domain F of the Hecke Eisenstein series G_k^{A,B} (at fixed level N) as the even weight k→∞, and characterize whether the zeros lie on the boundary arc and/or on specified vertical geodesics (e.g., for odd N on |z|=1 and on Re(z)=±1/(2j) for j=1,…,(N−1)/2).

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Background

The paper focuses on an alternative basis E_k{a,b}, for which geodesic clustering and limits are proved. The classical Hecke Eisenstein series G_k{A,B} are related by a finite Fourier transform, but their zero geometry in the large-weight limit is not established here.

Numerical experiments suggest a different structured limit set for G_k{A,B}, motivating a precise determination.

References

We list some open problems. Find the Hausdorff limit of zeros of the `Hecke' Eisenstein series G_k{A,B} in F as the (even) weight increases. Experiments suggest that these zeros are all located on the boundary arc, or on specific vertical lines defined by fixed rational numbers depending on N. For example, for odd N, the zeros appear to be on |z=1| and Re(z)=\pm 1/(2j) for j=1,\dots,(N-1)/2.

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