Numerical evaluation of spherical GJMS determinants for even dimensions

Abstract

The functional determinants of the GJMS scalar operators, P_{2k}, on even-dimensional spheres are computed via Barnes multiple gamma functions relying on the numerical availability of the digamma function. For the critical k=d/2 case, it is necessary to calculate the Stirling moduli. The multiplicative anomalies are given as odd polynomials in $k$ and it is emphasised that that the Dirichlet--to--Robin factorisation, P_{2l+1}, gives the same results as P_{2k} if k=l+1/2.The results are presented as graphs and show a series of extrema in the effective action as k is varied in the reals. For odd dimensions these extrema occur at integer k.