Euler Product Asymptotics for L-functions of Elliptic Curves

Let E/ Q be an elliptic curve and for each prime p, and let Nₚ denote the number of points of E modulo p. The original version of the Birch and Swinnerton-Dyer conjecture asserts that ∏ p ≤ x Nₚp ~ C (log x)rank(E( Q)) as x → ∞. Goldfeld (1982) showed that this conjecture implies both the Riemann Hypothesis for $L(E, s)$ and the modern formulation of the conjecture, that is, ord ₛ₌₁ L(E, s)= rank(E( Q)). In this paper, we prove that if we let r=ord ₛ₌₁L(E, s), then under the assumption of the Riemann Hypothesis for $L(E, s)$, we have that ∏ p ≤ x Nₚp ~ C (log x)ʳ for all x outside a set of finite logarithmic measure. As corollaries, we not only recover Goldfeld’s result, but also prove a result in the direction of the converse. Our method of proof is based on establishing the asymptotic behaviour of partial Euler products of $L(E, s)$ in the right-half of the critical strip.