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Zeros of Dirichlet \(L\)-functions. (English) Zbl 0771.11033
This interesting paper deals with the problem of the frequency with which \(L(s,\chi)\neq 0\), where \({1\over 2}\leq s<1\) and \(\chi\) is a Dirichlet character. The main results are as follows. Let \(q\) be a sufficiently large prime. Then, for a positive proportion of the characters \(\chi\pmod q\), one has that (i) \(L({1\over 2},\chi)\neq 0\) and (ii) there are no real zeros of \(L(s,\chi)\) with \({1\over 2}+c/\log q\leq\sigma<1\).
The new idea introduced in the paper is to count the desired characters directly, without the intermediary of moments of \(L\)-functions, which was the method used in previous work.
Reviewer: A.Perelli (Genova)

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