Zeta functions: formulas and applications

The existence conditions of the zeta function of a pseudodifferential operator and the definition of determinant thereby obtained are reviewed, as well as the concept of multiplicative anomaly associated with the determinant and its calculation by means of the Wodzicki residue. Exponentially fast convergent formulas – valid in the whole of the complex plane and yielding the pole positions and residua – that extend the ones by Chowla and Selberg for the Epstein zeta function (quadratic form) and by Barnes (affine form) are then given. After briefly recalling the zeta function regularization procedure in quantum field theory, some applications of these expressions in physics are described.