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Abstract

Let $\chi _D$ be the Dirichlet character associated to $\mathbb {Q}(\sqrt {D})$ where $D \lt 0$ is a fundamental discriminant. Improving Granville–Stark’s 2000 result, we show that \[ \frac {L’}{L}(1,\chi _D) = \frac {1}{6} \mathop {\rm ht}(j(\tau _D)) - \frac {1}{2}\log |D| + C + o_{D\to -\infty }(1), \] where $\tau _D = \frac 12(-\delta +\sqrt {D})$ for $D \equiv \delta \pmod {4}$ and $j(\cdot )$ is the $j$-invariant function with $C = -1.057770\ldots .$ Assuming the “uniform” $abc$-conjecture for number fields, we deduce that $L(\beta ,\chi _D)\ne 0$ with $\beta \geq 1 - \frac {\sqrt {5}\varphi + o(1)}{\log |D|}$ where $\varphi = (1+\sqrt {5})/2$, which we moreover improve for smooth $D$.