Basic Analysis of Regularized Series and Products

Analytic number theory and part of the spectral theory of§operators (differential, pseudo-differential, elliptic,§etc.) are being merged under amore general analytic theory§of regularized products of certain sequences satisfying a§few basic axioms. The most basic examples consist of the§sequence of natural numbers, the sequence of zeros with§positive imaginary part of the Riemann zeta function, and§the sequence of eigenvalues, say of a positive Laplacian on§a compact or certain cases of non-compact manifolds. The§resulting theory is applicable to ergodic theory and§dynamical systems; to the zeta and L-functions of number§theory or representation theory and modular forms; to§Selberg-like zeta functions; andto the theory of§regularized determinants familiar in physics and other parts§of mathematics. Aside from presenting a systematic account§of widely scattered results, the theory also provides new§results. One part of the theory deals with complex analytic§properties, and another part deals with Fourier analysis.§Typical examples are given. This LNM provides basic results§which are and will be used in further papers, starting with§a general formulation of Cram r's theorem and explicit§formulas. The exposition is self-contained (except for§far-reaching examples), requiring only standard knowledge of§analysis.