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.