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A note on the least prime in an arithmetic progression. (English) Zbl 0855.11045
For coprime numbers \(k\), \(\ell\), let \(p(k, \ell)\) denote the least prime \(p\equiv \ell \pmod k\), and let \(P(k)\) denote the largest of these \(\varphi (k)\) primes. Yu. V. Linnik proved that there is a constant \(L\) such that \(P(k) \ll k^L\), and recently D. R. Heath-Brown [Proc. Lond. Math. Soc., III. Ser. 64, 265-338 (1992; Zbl 0739.11033)] showed that \(L= 5.5\) is an admissible value. The value for \(P(k)\) is at least as big as the \(\varphi (k)\)-th prime, which has the asymptotic value \(\varphi (k)\log k\). Thus \[ \alpha= \liminf_{k\to \infty} {{P(k)} \over {\varphi (k) \log k}} \geq 1, \] and it is conjectured that \(P(k)\gg \varphi (k) \log^2 k\), so that the expected value for \(\alpha\) is \(\infty\). C. Pomerance [J. Number Theory 12, 218-223 (1980; Zbl 0436.10020)] proved that \(\alpha \geq e^\gamma= 1.78107 \dots\), where \(\gamma\) is Euler’s constant. The author gives an elaboration of the argument by considering the average distribution of primes in special arithmetic progressions, thereby improving the result to \(\alpha\geq e^\gamma+ 1\).

11N13 Primes in congruence classes