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The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta function $\zeta(z)$ lie on the critical line $\Re(z)=1/2$.

The MathWorld page on this topic mentions that the hypothesis has been verified for the first ten trillion $(10^{13})$ zeros. Unfortunaly I have never seen an explicit proof that the first non-trivial zero $$\rho_1\approx0.5000000000...+i\,14.1347251417...$$ lies exactly on the critical line. Could you please show me a proof of this?

OlegK
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  • Peter Sarnak once gave a brief mention in a talk, a long time ago. You place a rectangle around the apparent zero and do a certain integral around that curve. It would probably need to be the reciprocal of $\zeta$ because you need a pole to get a nonzero integral. You do the integral, numerically i think, and show that there is one pole inside the rectangle. By the reflection symmetry (includes an auxiliary function), that must lie on the critical strip. – Will Jagy Aug 18 '15 at 17:12
  • Specifically it looks like a combination of the answers in the possible duplicate should address your question. – user21820 Aug 18 '15 at 17:13
  • @user21820 , it has been so long, what would be techniques for showing that a specific contour in $\mathbb C$ contains exactly one pole of a meromorphic function, rather than two or more? – Will Jagy Aug 18 '15 at 17:17
  • @WillJagy: You're asking me? Sorry I don't know anything much about the Riemann hypothesis... I just thought the answers might satisfy the asker, not myself. =) – user21820 Aug 18 '15 at 17:19
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    I see that you are want the proof of RH with other way of asking !!!!!! – zeraoulia rafik Aug 18 '15 at 17:35
  • https://en.wikipedia.org/wiki/Argument_principle "The argument principle can be used to efficiently locate zeros or poles of meromorphic functions on a computer. Even with rounding errors, the expression {1\over 2\pi i}\oint_{C} {f'(z) \over f(z)}, dz will yield results close to an integer; by determining these integers for different contours C one can obtain information about the location of the zeros and poles. Numerical tests of RH use this technique to get an upper bound for the number of zeros of Riemann's \xi(s) function inside a rectangle intersecting the critical line." – Will Jagy Aug 18 '15 at 17:52

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