Analytic number theory and part of the spectral theory ofoperators (differential, pseudo-differential, elliptic,etc.) are being merged under amore general analytic theoryof regularized products of certain sequences satisfying afew basic axioms. The most basic examples consist of thesequence of natural numbers, the sequence of zeros withpositive imaginary part of the Riemann zeta function, andthe sequence of eigenvalues, say of a positive Laplacian ona compact or certain cases of non-compact manifolds. Theresulting theory is applicable to ergodic theory anddynamical systems; to the zeta and L-functions of numbertheory or representation theory and modular forms; toSelberg-like zeta functions; andto the theory ofregularized determinants familiar in physics and other partsof mathematics. Aside from presenting a systematic accountof widely scattered results, the theory also provides newresults. One part of the theory deals with complex analyticproperties, and another part deals with Fourier analysis.Typical examples are given. This LNM provides basic resultswhich are and will be used in further papers, starting witha general formulation of Cram r's theorem and explicitformulas. The exposition is self-contained (except forfar-reaching examples), requiring only standard knowledge ofanalysis.