Spectral determinants and zeta regularization

Sum negative powers only over strictly positive eigenvalues, retaining their multiplicities. Weyl's law gives absolute convergence when the real part of the complex exponent exceeds half the dimension. The sum initially defines an analytic function there, not near zero in general. Analytic continuation is therefore a mathematical step, not permission to insert zero into a divergent series. If the operator has a nontrivial kernel, its dimension must be recorded separately rather than assigned an infinite or vanishing summand.

For the smooth compact scalar settings stated here, heat asymptotics provide a meromorphic continuation that is regular at zero. The regularized positive-spectrum determinant is the exponential of minus its derivative at zero. It is not the limit of uncorrected partial products, and it is not the ordinary determinant of an operator with a kernel. Singular spaces, unusual boundary problems, or operators with logarithmic heat terms need separate continuation analysis; regularity at zero should not be presumed for every operator someone calls Laplace-like.

Integrating the heat weights against time to the exponent s minus one produces a gamma factor times the spectral powers. The full heat trace must have the kernel dimension subtracted, otherwise its constant large-time tail prevents convergence. At small time, subtract finitely many known asymptotic terms inside a split integral and integrate those terms explicitly. Their elementary poles continue the zeta function beyond its initial half-plane, while the remainder converges in a larger region.

This construction connects local heat coefficients with global spectral determinants, but the determinant is not determined by only the first two heat coefficients. The whole positive-time trace contributes to the continued value. The gamma factor also has a zero in its reciprocal at zero, which helps account for regularity in the standard smooth setting. A constant term in a reduced heat expansion already includes the kernel subtraction; adding or removing it a second time changes the answer.

Multiplying every positive eigenvalue by a positive constant c multiplies the zeta function by $c^{-s}$. Differentiating at zero shows that the determinant scales by $c^{\zeta_\Delta(0)}$, an exponent supplied by continuation rather than an infinite count of eigenvalues. Metric length scaling replaces the Laplacian by its reciprocal-square multiple. The resulting determinant power can therefore depend on the operator, dimension, and removed kernel; it is not generally the finite-dimensional determinant rule with an unspecified infinite exponent.

Intervals and circles make these distinctions explicit because their zeta functions reduce to the Riemann zeta function. Use its verified values at zero and its derivative at zero, rather than attempting to multiply integer squares. A circle has double positive frequencies and one excluded zero mode; a Dirichlet interval has simple positive frequencies and no kernel. Those differences produce different determinant length powers, even though both examples are one-dimensional and locally flat.