Abstract
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.
Highlights
FIELD: Mathematics ARTICLE TYPE: Review paper Abstract: 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 + · · ·
Complex plane integrations and properties of the Gamma function will be used from the definition of the function to its analytic extension
This defines the zeta function, already known to Euler (Euler, 1738), (Euler, 1740), the properties of which were discovered by Riemann (Riemann, 1859) more than 100 years after Euler’s works
Summary
FIELD: Mathematics ARTICLE TYPE: Review paper Abstract: 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 + · · ·.
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