Zeta-regularization of arithmetic sequences

Is it possible to give a reasonable value to the infinite product 1 × 2 × 3 × · · · × n × · · · ? In other words, can we define some sort of convergence of the finite product 1 × 2 × 3 × · · · × n when n goes to infinity? One way is to relate this product to the R√iemann zeta function and to its analytic continuation. This approach leads to: 1 × 2 × 3 × · · · × n × · · · = 2π. More generally the “zeta-regularization” of an infinite product consists of introducing a related Dirichlet series and its analytic continuation at 0 (if it exists). We will survey some properties of this generalized product and allude to applications. Then we will give two families of possibly new examples: one unifies and generalizes known results for the zeta-regularization of the products of Fibonacci, balanced and Lucas-balanced numbers; the other studies the zeta-regularized products of values of classical arithmetic functions. Finally we ask for a possible zeta-regularity notion of complexity