Insights The Extended Riemann Hypothesis and Ramanujan’s Sum

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The discussion centers on the Riemann Hypothesis (RH) and the Extended Riemann Hypothesis (ERH), highlighting that RH asserts all non-trivial zeros of the Riemann zeta function lie on the critical line, while ERH extends this to L-functions of complex Dirichlet characters. The conversation emphasizes the significance of understanding key concepts such as the critical strip, trivial and non-trivial zeros, and group characters. The importance of these hypotheses is underscored by their implications in various mathematical fields, particularly in modern computer science. Participants express fascination with the Riemann Hypothesis, reflecting on its simplicity and profound impact. The discussion concludes with a nod to the ongoing efforts in proving these mathematical mysteries.
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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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There's something about the Riemann hypothesis that fascinates me. And this is coming from a guy who had to take calc 2, calc 3, and diff eq twice each to pass them. It seems like such a simple problem, yet has profound implications in many different areas of math. Best of luck to all those trying to solve it!
 
Drakkith said:
Best of luck to all those trying to solve it!
David Hilbert said:
If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?

"Mathematical Mysteries: The Beauty and Magic of Numbers". Book by Calvin C. Clawson, p. 258, 1999.
 
Keep at it mathematicians!

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It poses important moral dilemmas.

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