Spectral zeta functions in positive characteristic

Spectral zeta functions and zeta-regularized determinants encode the spectrum of an operator in analytic functions. Over the completion $\mathbb C_\infty$ of an algebraic closure of $\mathbb F_q((π))$, two features of characteristic $p$ break this classical picture. Complex powers have exponents that vary only continuously in $\mathbb Z_p$, not analytically. Scalar traces record eigenvalue multiplicities only modulo $p$. We develop a resolvent-based theory that treats these two obstructions separately. For the first, inverse Stieltjes duality turns suitable scalar resolvents into locally analytic spectral distributions on $\mathbb F_q[[π]]$. A single exponent of the Mellin transform already determines the distribution and the entire Mellin family. We characterize exactly which power series arise in this way, by a growth condition on Carlitz moments. The criterion depends on the support: bounded Taylor coefficients are not sufficient. For stationary digit measures, the transform is the unique fixed point of an explicit contraction. The first $q-1$ nonconstant moments recover the measure. For the second, we replace traces by the Frobenius defect $F/σ(F)$, where $σ$ raises coefficients to their $q$th powers. On power series with small coefficients, this map is a group automorphism that preserves Gauss distances, and a convergent infinite product inverts it. Polynomial defects produce entire functions. They characterize exactly the diagonal spectra that are finite unions of forward Frobenius orbits, counted with multiplicity. For the shifted Carlitz operator, the resulting Fredholm determinant is a normalized reciprocal of the Anderson--Thakur function, and its Newton polygon is determined exactly.