Real zeros of quadratic Dirichlet \(L\)-functions.

*(English)*Zbl 1042.11053It is generally believed that no Dirichlet \(L\)-function has a real positive zero, but even the possibility that every \(L\)-function of sufficiently large conductor has a real zero had not been excluded before the appearance of this paper. But now this possibility is eliminated, and the \(L\)-functions with the desired non-vanishing property correspond to certain quadratic characters. Let \(\chi _d(n)\) denote the real primitive character related to the fundamental discriminant \(d\). The main result of this highly interesting paper is that for all large \(x\) the number of odd positive square-free integers \(d\leq x\) such that \(L(\sigma , \chi _{-8d})>0\) for all \(0\leq \sigma \leq 1\) exceeds \(4x/(5\pi ^2)\). The point of the result is that it holds even off the point \(1/2\); an analogous result on the non-vanishing of \(L(1/2,\chi )\) was recently proved by K. Soundararajan [Ann. Math. (2) 152, 447–488 (2000; Zbl 0964.11034)], and in fact the present paper is methodically close to that one. In particular, mollification techniques and mean value estimates for quadratic character sums play important roles as before.

Reviewer: Matti Jutila (Turku)