Nonvanishing of quadratic Dirichlet $L$-functions at $s=\frac{1}{2}$

K. Soundararajan

Review from Zentralblatt MATH:

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 $L(1/2,\chi_{8d})\ne 0$ for at least $15/16+ o(1)$ of all relevant values of $d\le x$, as $x\to\infty$.

Previously one only knew, from work of {\it M. Jutila} [Analysis 1, 149-161 (1981; 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 $$\sum_{d\le x}\mu(2d)^2 L(\textstyle{1\over 2}, \chi_{8d})^k= xP(\log x)+ O(x^{\theta+ \varepsilon}),$$ 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.

Reviewed by Roger Heath-Brown

Keywords: Dirichlet $L$-function; real character; quadratic character; real zero; critical value; nonvanishing theorem; mollifier

Classification (MSC2000): 11M20

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