## Nonvanishing of quadratic Dirichlet $$L$$-functions at $$s=\frac{1}{2}$$.(English)Zbl 0964.11034

Let $$d$$ be an odd, positive, square-free integer, and let $$\chi_{8d} (n)$$ be the real primitive character of conductor $$8d$$, given by the Jacobi symbol $$(8d/n)$$. Then it is shown that $$L(1/2,\chi_{8d})\neq 0$$ for at least $$15/16+ o(1)$$ of all relevant values of $$d\leq x$$, as $$x\to\infty$$.
Previously one only knew, from work of M. Jutila [Analysis 1, 149-161 (1981; Zbl 0485.10029)], that $$\gg x(\log x)^{-1}$$ values of $$d\leq x$$ produced non-vanishing values of $$L(1/2,\chi_{8d})$$.
For the proof one requires asymptotic formulae for the mollified moments $\sum_{d\leq x}\mu (2d)^2|L(\textstyle {1\over 2},\chi_{8d}) M(d)|^k,$ for $$k=1$$ and 2, where $M(d)= \sum_{m\leq M}w_mm^{-1/2} \chi_{8d}(m),$ is a mollifier of length $$M=x^{1/2-\varepsilon}$$, and $$w_m$$ are suitable weights. The error terms for these moments are handled using the reviewer’s large sieve estimate for real characters [Acta Arith. 72, 235-275 (1995; Zbl 0828.11040)], but the principal difficulty lies in the evaluation of the main terms. Here there is a non-zero contribution from the off-diagonal terms, in the case of the second moment. It is also shown that $\sum_{d\leq x}\mu(2d)^2 L(\textstyle{1\over 2}, \chi_{8d})^k= xP(\log x)+ O(x^{\theta+ \varepsilon}),$ for $$k=2$$ or 3, where $$P$$ is a polynomial of degree 3 or 6 respectively, and $$\theta=5/6$$ for $$k=2$$, and $$\theta= 11/12$$ for $$k=3$$.
The proofs are difficult, and, for the nonvanishing theorem, rely on some apparently fortuitous cancellations. However the results represent a very considerable advance on what was previously known.

### MSC:

 11M20 Real zeros of $$L(s, \chi)$$; results on $$L(1, \chi)$$

### Citations:

Zbl 0485.10029; Zbl 0828.11040
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