Simple evaluations of error terms in zeta-related identities

Determine whether the quantities ζ₂(a, b), ∫_{0}^{π/2} z^{2m} cot(z) dz, and ψ^{(2n−1)}(1/4) admit simple evaluations in terms of common mathematical constants, and, if so, derive explicit closed-form expressions.

Background

In surveying known identities and recurrences for odd zeta values and related constants, the author notes that several associated ‘error terms’—Euler double zeta sums ζ₂(a, b), certain cotangent integrals ∫_{0}^{π/2} z^{2m} cot(z) dz, and specific polygamma values ψ^{(2n−1)}(1/4)—do not have known simple closed forms in terms of standard constants.

This observation underscores a broader unresolved issue within the literature: even when structural recurrences or identities are available, key auxiliary quantities still lack closed-form evaluations, limiting further simplification and analytic understanding.