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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 \(L(1, \chi)\)
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