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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 $$\alpha\geq e^\gamma+ 1$$.

MSC:
 11N13 Primes in congruence classes