See Deninger’s articles.
Zeta regularisation: Book by Elizalde in L-functions folder
http://mathoverflow.net/questions/183/zeta-function-regularization-of-determinants-and-traces
http://mathoverflow.net/questions/94512/understanding-zeta-function-regularization
arXiv:1303.7028 Zeta regularized products, Riemann zeta zeros and prime number spectra
nLab page on Regularization