  The conformal anomalies and functional determinants of the Branson--GJMS
operators, P_{2k}, on the d-dimensional sphere are evaluated in explicit terms
for any d and k such that k < d/2+1 (if d is even). The determinants are given
in terms of multiple gamma functions and a rational multiplicative anomaly,
which vanishes for odd d. Taking the mode system on the sphere as the union of
Neumann and Dirichlet ones on the hemisphere is a basic part of the method and
leads to a heuristic explanation of the non--existence of `super--critical'
operators, 2k>d for even d. Significant use is made of the Barnes zeta
function. The results are given in terms of ratios of determinants of operators
on a (d+1)-dimensional bulk dual sphere. For odd dimensions, the log
determinant is written in terms of multiple sine functions and agreement is
found with holographic computations, yielding an integral over a Plancherel
measure. The N-D determinant ratio is also found explicitly for even
dimensions. Ehrhart polynomials are encountered.
