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Recent work on Euclidean quantum gravity on the four-ball has provedregularity at the origin of the generalized zeta-function built fromeigenvalues for metric and ghost modes, when diffeomorphism-invariant boundaryconditions are imposed in the de Donder gauge. The hardest part of the analysisinvolves one of the four sectors for scalar-type perturbations, the eigenvaluesof which are obtained by squaring up roots of a linear combination of Besselfunctions of integer adjacent orders, with a coefficient of linear combinationdepending on the unknown roots. This paper obtains, first, approximate analyticformulae for such roots for all values of the order of Bessel functions. Forthis purpose, both the descending series for Bessel functions and their uniformasymptotic expansion at large order are used. The resulting generalizedzeta-function is also built, and another check of regularity at the origin isobtained. For the first time in the literature on quantum gravity on manifoldswith boundary, a vanishing one-loop wave function of the Universe is found inthe limit of small three-geometry, which suggests a quantum avoidance of thecosmological singularity driven by full diffeomorphism invariance of theboundary-value problem for one-loop quantum theory.Comment: 21 Pages, Latex and .eps files with JHEP3 style. The discussion in  Section 5 has been improved, and Ref. 26 has been adde

A non-singular one-loop wave function of the universe from a new eigenvalue asymptotics in quantum gravity