Скачать или смотреть Products of Dirichlet Series and Zeta Regularization: Minicourse on the Dirichlet series

In this lecture, we explore the product structure of Dirichlet series, extending our understanding of the zeta function and its multiplicative properties. We introduce the Dirichlet eta function and discuss its role in analytic continuations of Dirichlet series.

A key focus of this lecture is zeta regularization, a powerful technique used in physics—particularly in quantum field theory (QFT)—where divergent series and infinite sums appear naturally. Applications include:

Casimir effect – regularizing vacuum energy calculations
Quantum statistical mechanics – linking partition functions to zeta functions
Path integrals in QFT – handling infinite mode sums in functional determinants
By manipulating Dirichlet series products and applying zeta regularization techniques, we gain deeper insights into both mathematical structures and their physical applications. This lecture builds upon our previous discussion of the zeta function and primes the groundwork for connections to spectral functions and partition functions in upcoming lectures.

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