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Riemann Hypothesis and Ramanujan’s Sum Explanation

RH:All non-trivial zeros of the Riemannian zeta-function lie on the critical line.

ERH:All zeros of L-functions to complex Dirichlet characters of finite cyclic groups within the critical strip lie on the critical line.

Related Article:The History and Importance of the Riemann Hypothesis

The goal of this article is to provide the definitions and theorems that are necessary to understand these two Riemann hypotheses, i.e. why is a strip in the complex plane called critical, what are trivial and non-trivial zeros, and what is a group character, etc. We will gather a couple of facts around the functions involved, in particular several functional equations.

The extended Riemann hypothesis is a generalization of the Riemann hypothesis that became important when modern computer science began to use...

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