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Honestly, I have to say that I have hardly any experience in number theory. That's maybe one additional reason why the Riemann hypothesis has such a "mystic" appearance for me. You always hear or read that it's basically "the" problem to solve in mathematics. But you always just read (as a non-mathematician) that it "has something to do with the distribution of primes".

But how do the (nontrivial) roots of $\zeta(s)$ "connect" to the distribution of primes? What's the point that makes these roots so crucial?

Thanks in advance for any answer!

EDIT: Adapted the questions, as the original ones were to broad.

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closed as too broad by user149792, user147263, colormegone, Strants, Andrés E. Caicedo Sep 5 '15 at 12:54

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ You ask several questions, and it seems that a short essay could not give a proper answer to all of them. The question may be too broad, but I would like to see an answer. $\endgroup$ – Joonas Ilmavirta Sep 4 '15 at 20:20
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    $\begingroup$ To phrase your question directly: what are the consequence of the Riemann Hypothesis? $\endgroup$ – Asaf Karagila Sep 4 '15 at 20:36
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    $\begingroup$ @quid: No, it would be much much cooler if it were proved to be equivalent to something like Con(ZFC). $\endgroup$ – Asaf Karagila Sep 4 '15 at 20:39
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    $\begingroup$ RH itself implies a tighter bound on the distribution of primes - indeed the zeta zeros act as "Fourier frequencies" for the prime counting function, and different real parts for those zeros would mean different frequencies have different orders of magnitude of contribution to the count. Whether or not a proof of RH would shed additional light on the primes, we don't know. We'd have to see the proof to know! $\endgroup$ – whacka Sep 4 '15 at 20:42
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    $\begingroup$ @Kyle: Au contraire. If RH is provable and equivalent to Con(ZFC) then there is a very deep foundational issue. And it should be fascinating to come up with a better foundation and salvage what we know about mathematics today. $\endgroup$ – Asaf Karagila Sep 4 '15 at 21:31