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On the least prime in an arithmetic progression and estimates for the zeros of Dirichlet L-functions. (English) Zbl 1248.11067

Yu. V. Linnik [On the least prime in an arithmetic progression. I: The basic theorem”, Mat. Sb., N. Ser. 15(57), 139–178 (1944; Zbl 0063.03584); “On the least prime in an arithmetic progression. II: The Deuring-Heilbronn theorem”, Mat. Sb., N. Ser. 15(57), 347–368 (1944; Zbl 0063.03585)] proved that there exists an effectively computable constant \(L\) such that the least prime in an arithmetic progression \(a\) mod \(q\) with \((a,q)=1\) is \(O\left(q^L\right)\). C. D. Pan [“On the least prime in an arithmetical progression”, Sci. Record, n. Ser. 1, 311–313 (1957; Zbl 0083.26203)] established in 1957 that \(L=10000\) is admissible. Since then, a lot of research has been done on Linnik’s constant \(L\). The latest record \(L=5.5\) is due to [D. R. Heath-Brown, “Zero-free regions for Dirichlet \(L\)-functions and the least prime in an arithmetic progression”, Proc. Lond. Math. Soc., III. Ser. 64, No. 2, 265–338 (1992; Zbl 0739.11033)]. In the said paper, Heath-Brown makes nine suggestions of small improvements. By carrying out four of them, the author manages to lower the value of Linnik’s constant to \(L=5.18\). This is mainly achieved by improving several of Heath-Brown’s results concerning zero-free regions and zero-density estimates for Dirichlet \(L\)-functions, which requires a large amount of technical work.

MSC:

11N13 Primes in congruence classes
11M06 \(\zeta (s)\) and \(L(s, \chi)\)