The average analytic rank of elliptic curves

@article{HeathBrown2003TheAA,
  title={The average analytic rank of elliptic curves},
  author={D. R. Heath-Brown},
  journal={Duke Mathematical Journal},
  year={2003},
  volume={122},
  pages={591-623}
}
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 [2]. We also show that the average within any family of quadratic twists is at most 3/2, improving a result of Goldfeld [3]. A third result concerns the density of curves with analytic rank at least R, and shows that the proportion of such… 
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