  Recent work on Euclidean quantum gravity on the four-ball has proved
regularity at the origin of the generalized zeta-function built from
eigenvalues for metric and ghost modes, when diffeomorphism-invariant boundary
conditions are imposed in the de Donder gauge. The hardest part of the analysis
involves one of the four sectors for scalar-type perturbations, the eigenvalues
of which are obtained by squaring up roots of a linear combination of Bessel
functions of integer adjacent orders, with a coefficient of linear combination
depending on the unknown roots. This paper obtains, first, approximate analytic
formulae for such roots for all values of the order of Bessel functions. For
this purpose, both the descending series for Bessel functions and their uniform
asymptotic expansion at large order are used. The resulting generalized
zeta-function is also built, and another check of regularity at the origin is
obtained. For the first time in the literature on quantum gravity on manifolds
with boundary, a vanishing one-loop wave function of the Universe is found in
the limit of small three-geometry, which suggests a quantum avoidance of the
cosmological singularity driven by full diffeomorphism invariance of the
boundary-value problem for one-loop quantum theory.
