Determine whether exact count recovery can use only zero contributions

Determine whether the exact Perron recovery formula for the almost-prime counting function can be expressed through a convergent contribution from the nonreal zeta-zero slits alone, without requiring the outer-boundary term or an unseparated combined limit.

Background

The paper deforms a finite Perron integral for the Dirichlet series H_k(s) into a real-cut contribution, denoted _k(x), together with contributions from nonreal slits associated with scaled nontrivial zeros of the Riemann zeta function and an additional outer-boundary term. The resulting identity recovers the symmetrized almost-prime count N_k^*(x) only through the combined limit of these complementary contour terms.

The authors explicitly refrain from claiming that the nonreal-zero contribution and the outer-boundary contribution converge separately, or that the outer-boundary term vanishes. The unresolved issue is therefore whether an exact convergent formula can be obtained from the zero-associated contributions alone, which would eliminate the need for the outer-boundary contribution in the recovery of N_k^*(x).