Explicit formulae and the Lang-Trotter conjecture
Let \(E\) be an elliptic curve defined over \(\mathbb Q\) of conductor \(N\). Define \[ \pi_ E(x)=\#\{p<x: p\nmid N\text{ and } N_ p=p+1\} \] where \(N_ p\) is the number of points of the reduction of \(E \mod p\) defined over the field of \(p\) elements. Let \(L_ k(s)\) be the ``\(k\)-th symmetric power'' of the \(L\)-series attached to \(E\). Assuming analytic continuation, functional equation and Riemann hypothesis, the author studies by analytic methods the integral \[ \frac 1{2\pi i}\int^{c+iT}_{c- iT}(L'/L)(s) x^ s \,\frac{ds}{s} \] and deduces for example that if \(E\) does not have complex multiplication, then \[ \pi_ E(x)\ll x^{3/4}(\log Nx)^{1/2}. \]
scientific article; zbMATH DE number 3941675
scientific article