Bertrand's postulate for primes in arithmetical progressions
The author considers the following generalization of Bertrand's postulate. Define, for integers \(m \geq 1\), \(d \geq 2\), and a real \(z>1\) \[ \begin{multlined} B_ m (z,d)=\lim \inf \{c:\text{ For every } x \geq c \text{ the interval } (x,zx) \text{ contains } \\ \text{at least } m \text{ primes } \equiv a \pmod d \}.\end{multlined} \] Using elementary methods of \textit{G. Ricci} [Boll. Un. Mat. Ital. 13, 7-17 (1934; Zbl 0008.29602)] and \textit{P. Erdős} [Math. Z. 39, 473-491 (1935; Zbl 0010.29303)] he derives explicit upper bounds for \(B_ m(z,d)\) in the case \[ \sigma (d)=\sum_{p<d, p \nmid d} {1 \over p}<1. \] A complete list of these numbers \(d\) is given, the largest one is \(d=840\).
scientific article; zbMATH DE number 474523
scientific article