Some analytic continuations of the Barnes zeta function in two and higher dimensions

Formulas for the analytic continuation of the Barnes zeta function, and some affine extensions thereof, in two and more dimensions, are constructed. The expressions are used to deal with determinants of multidimensional harmonic oscillators. An example is therewith obtained of the multiplicative anomaly (or defect), associated with the most common definition (due to Ray and Singer [D.B. Ray, Reidemeister torsion and the Laplacian on lens spaces, Adv. Math. 4 (1970) 109–126; D.B. Ray, I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971) 145–201; D.B. Ray, I.M. Singer, Analytic torsion for complex manifolds, Ann. Math. 98 (1973) 154–177]) of determinant of a pseudodifferential operator admitting a zeta function [M. Kontsevich, S. Vishik, Geometry of determinants of elliptic operators, in: Functional Analysis on the Eve of the 21st Century, vol. 1, 1995, pp. 173–197].