Gene

Quantum mechanics

Curvature

Pure mathematics

Mathematical physics

Laplace operator

Hilbert space

Arithmetic zeta function

Biochemistry

Computer science

Constant curvature

Riemann zeta function

Spectrum (functional analysis)

Physics

Riemann hypothesis

Mathematics

Transcription factor

Mathematical analysis

Artificial intelligence

Geometry

Repressor

Chemistry

Operator (biology)

Class (philosophy)

Spectral theory

Advanced studies in pure mathematics

This article gives a survey of various generalizations of Riemann's $\zeta$-function, associated with operator spectra and which may be generically called spectral zeta functions. Areas of application include Riemannian geometry (the spectrum of the Laplacian) and quantum mechanics. We review one example of each class in concrete detail: the Laplacian on a compact surface of constant negative curvature, and the Schrödinger operator on the real line with a homogeneous potential $q^{2M}$ ($M$ a positive integer).

Spectral Zeta Functions