From the Birch and Swinnerton-Dyer conjecture to Nagao’s conjecture

Let E E be an elliptic curve over Q Q with discriminant Δ E Δ _E . For primes p p of good reduction, let N p N_p be the number of points modulo p p and write N p = p + 1 − a p N_p=p+1-a_p . In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies lim x → ∞ 1 log ⁡ x ∑ p ≤ x p ∤ Δ E a p log ⁡ p p = − r + 1 2 , {equation*} lim x→ ∞1/log x∑ _{ {p≤ x\\ p Δ _{E