Let p ≡ 4, 7 mod 9 be a rational prime number such that 3 mod p is not a cube. In this paper, we prove the 3-part of | III( E p ) |⋅| III( E 3 p 2 ) | is as predicted by the Birch and Swinnerton-Dyer ...
We prove an explicit formula, analogous to the classical explicit formula for ψ(x), for the Cesáro–Riesz mean of any order k > 0 of the number of representations of n as a sum of two primes. Our ...