From the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture.

Let E be an elliptic curve over \mathbb {Q} with discriminant \Delta _E. For primes p of good reduction, let N_p be the number of points modulo p and write N_p=p+1-a_p. In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies \begin{equation*} \lim _{x\to \infty }\frac {1}{\log x}\sum _{\substack {p\leq x\\ p\nmid \Delta _{E}}}\frac {a_p\log p}{p}=-r+\frac {1}{2}, \end{equation*} where r is the order of the zero of the L-function L_{E}(s) of E at s=1, which is predicted to be the Mordell-Weil rank of E(\mathbb {Q}). We show that if the above limit exits, then the limit equals -r+1/2. We also relate this to Nagao's conjecture. This paper also includes an appendix by Andrew V. Sutherland which gives evidence for the convergence of the above-mentioned limit. [ABSTRACT FROM AUTHOR]