zeta function and some of its properties

Introduction/purpose: some properties of the zeta function will be shown as well as its applications in calculus, in particular the golden nugget formula for the value of the infinite sum 1 + 2 + 3 + · · · . some applications in physics will also be mentioned. methods: complex plane integrations and properties of the gamma function will be used from the definition of the function to its analytic extension. results: from the original definition of the z(s) function valid for s > 1 a meromorphic function is obtained on the whole complex plane with a simple pole in s = 1. conclusion: the relevance of the zeta function cannot be overstated, ranging from the infinite series to the number theory, regularization in theoretical physics, the casimir force, and many other fields.