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Zbl 0771.11033
Balasubramanian, R.; Murty, V.Kumar
Zeros of Dirichlet $L$-functions. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 25, No. 5, 567-615 (1992). ISSN 0012-9593

This interesting paper deals with the problem of the frequency with which $L(s,\chi)\ne 0$, 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) $L({1\over 2},\chi)\ne 0$ and (ii) there are no real zeros of $L(s,\chi)$ with ${1\over 2}+c/\log q\leq\sigma<1$.\par 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.
[A.Perelli (Genova)]

MSC 2000:: *11M20 Real zeros of L(s,chi)

Keywords: real zeros; Dirichlet $L$-functions

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