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Has it been ruled out that the Riemann hypothesis fails, but fails only for finite number of zeros?

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No. (Comments must be at least 15 characters.) –  stopple Dec 9 '11 at 0:03
A related post (mathoverflow.net/questions/50186/…) –  user17762 Dec 9 '11 at 0:31
@stopple, one trick is to append a dollar sign, a bunch of left and right brackets, and a dollar sign. –  Gerry Myerson Dec 9 '11 at 8:31
See?${}{}{}{}{}$ –  Gerry Myerson Dec 9 '11 at 8:31
Cool!${{{{{}}}}}$ –  stopple Dec 10 '11 at 21:53

1 Answer 1

Actually if The RH fails for a finite number of zeros, then it fails for all one can see this from Weil-Bombieri explicit formulae, if RH fails, then following Weil's "Sur les "formules explicites" de la theorie des nombres premiers" or Bombieri's "Remarks on Weil’s quadratic functional in the theory of prime numbers", one can always construct a "slowly varying" test function which will violate the STRONG positivity condition of Weil-Bombieri. So this positivity condition is strong enough to detect a single zero off the critical line !

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