Wall-sum evaluations for exceptional root systems E6, E7, and E8

Evaluate the wall sums corresponding to the second-pole residues of the normalized Witten zeta functions for the exceptional root systems E6, E7, and E8.

Background

The paper establishes that, for every irreducible reduced crystallographic root system of rank at least two, the first distinct pole below the leading pole occurs at q2=(r-1)/(N-1), is simple, and has a residue expressed as a positive sum of marked projective wall periods. Exact evaluations of these wall sums are obtained for types F4 and D5 using chamber-orbit relations together with Dixon and Selberg integral evaluations. The corresponding wall sums for E6, E7, and E8 are explicitly identified as remaining problems, with no evaluation supplied in the paper.