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Zero-free regions for Dirichlet \(L\)-functions. (English) Zbl 0934.11042

In work on Linnik’s constant [Proc. Lond. Math. Soc. (3) 64, 265-338 (1992; Zbl 0739.11033)] the reviewer established zero-free regions for Dirichlet \(L\)-functions to modulus \(q\), restricted to the region \(|t|\leq 1\). In the present paper arbitrary values of \(t\) are examined. It is proved that the region \[ \sigma\geq 1- \frac{0.239} {\log q(2+T)}, \qquad |t|\leq T \] contains at most an exceptional zero of \(\prod_{\chi\pmod q}L(s,\chi)\). Moreover the region \[ \sigma\geq 1- \frac{0.517} {\log q(2+T)}, \qquad |t|\leq T \] contains at most two zeros.
The proofs follow the reviewer’s methods. However, in place of Burgess’ bounds for \(L\)-functions (which are only efficient for small \(t\)), the author uses a hybrid bound \[ L(\tfrac 12+it,\chi)\ll (gT)^{3/16+ \varepsilon} \qquad (|t|\leq T,\;T\geq 1), \] due also to the reviewer [Q. J. Math., Oxf. II. Ser. 31, 157-167 (1980; Zbl 0427.10025)]. This is weaker than the Burgess result for \(|t|\leq 1\), so that the numerical constants 0.239 and 0.517 above are not as large as the corresponding ones for small \(t\).

MSC:

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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