Generalize the Riemann Hypothesis to elliptic curves and prove or disprove its validity in this context.
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The Riemann Hypothesis, formulated by Bernhard Riemann in 1859, is a conjecture regarding the distribution of prime numbers. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line, which has a real part of 1/2.
The connection between the Riemann Hypothesis and elliptic curves comes through the connection with the Birch and Swinnerton-Dyer Conjecture (BSD conjecture). The BSD conjecture is a major open problem in mathematics that relates the behavior of elliptic curves with the behavior of the Riemann zeta function.
In particular, the BSD conjecture states that there is a deep connection between the order of the zero at s=1 of the L-series associated with an elliptic curve and the rank of the elliptic curve. The L-series associated with an elliptic curve is defined using the Riemann zeta function and certain special functions related to the elliptic curve.
If the Riemann Hypothesis is true, then it would imply that the BSD conjecture is true for all elliptic curves. However, it is currently an unsolved problem, and neither the Riemann Hypothesis nor the BSD conjecture has been proven or disproven.
Therefore, we cannot generalize the Riemann Hypothesis to elliptic curves and prove or disprove its validity in this context at this time.
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