Sarnak–Strömbergsson Epstein conjecture below the convergence threshold

Determine the minimizers of the three-dimensional Epstein zeta problem for exponents below the convergence threshold s=3/2, thereby establishing the portion of the Sarnak–Strömbergsson Epstein conjecture not covered by the proved comparison for s>3/2.

Background

The paper proves that, among covolume-one three-dimensional lattices, the face-centred cubic lattice uniquely minimizes the Epstein zeta function for every s>3/2, where the defining Epstein series converges. It also treats the critical exponent s=3/2 through a regularized finite part rather than the divergent series.

The authors explicitly note that the Epstein conjecture of Sarnak and Strömbergsson extends to exponents below 3/2, but that this range is not covered by their theorem. Consequently, the conjectural minimization problem below the convergence threshold remains unresolved in the scope of the paper.