Under GRH the average analytic rank of y²=x³-dx over odd fourth-power-free d is at most 13/6, and at most 3/2 assuming a quartic Gauss-sum conjecture.
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Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$
Under GRH the average analytic rank of y²=x³-dx over odd fourth-power-free d is at most 13/6, and at most 3/2 assuming a quartic Gauss-sum conjecture.