The connection with zeros and primes?

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hi y'all after lurking a lot on this forum and searching for the answer I've got something to ask, if you get that all the non trivial zeros do lie on 1/2+ib then what? I've read a book on the riemann hypothesis but i really don't get the link between the zeros of zeta(s) and the prime counting function
 
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If you do prove it, then you get better bounds on errors in estimates in the prime counting function. Oh, and you get a million dollars.
 
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