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The average analytic rank of elliptic curves

D. R. Heath-Brown
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Abstract

All the results in this paper are conditional on the Riemann Hypothesis for the L-functions of elliptic curves. Under this assumption, we show that the average analytic rank of all elliptic curves over Q is at most 2, thereby improving a result of Brumer. We also show that the average within any family of quadratic twists is at most 3/2, improving a result of Goldfeld. A third result concerns the density of curves with analytic rank at least R, and shows that the proportion of such curves decreases faster than exponentially as R grows. The proofs depend on an analogue of Weil's ``explicit formula''.

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D. R. Heath-Brown. 2003-05-07. The average analytic rank of elliptic curves. https://arxiv.org/abs/math/0305114

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