Under RH and the Alternative Hypothesis, the densities P_{k/2} of zero pairs at normalized spacing k/2 satisfy P_{k/2} ~ P_0 - 1/2 (even k) and P_{k/2} ~ 3/2 - 2/(π^2 k^2) - P_0 (odd k), with 1 ≤ P_0 ≤ 3/2 - 2/π^2; a strengthened hypothesis yields P_0 = 1.
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The Alternative Hypothesis for Zeros of the Riemann Zeta-Function
Under RH and the Alternative Hypothesis, the densities P_{k/2} of zero pairs at normalized spacing k/2 satisfy P_{k/2} ~ P_0 - 1/2 (even k) and P_{k/2} ~ 3/2 - 2/(π^2 k^2) - P_0 (odd k), with 1 ≤ P_0 ≤ 3/2 - 2/π^2; a strengthened hypothesis yields P_0 = 1.