Zeta function and some of its properties - Fabiano, Nicola (Vojnotehnički glasnik / Military Technical Courier, 2020)
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 (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.
Keywords: Zeta function; analytic continuation; complex plane integration