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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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    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. – Joonas Ilmavirta Sep 04 '15 at 20:20
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    To phrase your question directly: what are the consequence of the Riemann Hypothesis? – Asaf Karagila Sep 04 '15 at 20:36
  • Actually, it 'd be much cooler it were disproved. – quid Sep 04 '15 at 20:36
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    @quid: No, it would be much much cooler if it were proved to be equivalent to something like Con(ZFC). – Asaf Karagila Sep 04 '15 at 20:39
  • I suppose there is also such a question in this site already but for the time being see http://mathoverflow.net/questions/17209/consequences-of-the-riemann-hypothesis // @Asaf but that be too cool then. :P – quid Sep 04 '15 at 20:41
  • https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequences_of_the_generalized_Riemann_hypothesis – Zubin Mukerjee Sep 04 '15 at 20:41
  • @quid I wouldn't say "cooler", but rather "interesting". I bet that first zero would be an incredibly important value, and it might even become a constant of sorts. – Zach466920 Sep 04 '15 at 20:42
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    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! – anon Sep 04 '15 at 20:42
  • @AsafKaragila: If the Riemann hypothesis is provable and equivalent to Con(ZFC), then I'm out of a job. I don't know if that sounds cool... – Kyle Gannon Sep 04 '15 at 20:47
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    @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. – Asaf Karagila Sep 04 '15 at 21:31
  • Wikipedia's article on RH has a section on its consequences. MathOverflow also has a question about RH's consequences. The first part of my answer here briefly explains the connection between zeta zeros and the distribution of primes. There are many pop-math books, textbooks, online documents and entire webpages devoted to RH. Search engines are your friend! – anon Sep 05 '15 at 16:17

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