| 41. | The extended Riemann hypothesis extends the Riemann hypothesis to all Dedekind zeta functions of algebraic number fields.
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| 42. | The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis.
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| 43. | So far, every single one of them has failed to provide a counterexample to the Riemann hypothesis.
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| 44. | Somewhat ironically, Li himself released a purported proof of the Riemann Hypothesis in the arXiv in July 2008.
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| 45. | Assuming the generalized Riemann hypothesis, R . C . Vaughan have shown that there is the further improvement
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| 46. | Leszek Kaniecki showed every odd integer is a sum of at most five primes, under the Riemann Hypothesis.
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| 47. | See the article on the Mertens conjecture for more information about the connection between and the Riemann hypothesis.
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| 48. | The Hooley published a conditional proof for the second conjecture, assuming certain cases of the Generalized Riemann hypothesis.
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| 49. | He formulated and proved a hypothesis that is analogous to it, but did not create the Riemann hypothesis itself.
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| 50. | The Riemann hypothesis is equivalent to several statements showing that the terms of the Farey sequence are fairly regular.
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