A note on Chebyshev’s theorem

We revisit a classical theorem of Chebyshev about distribution of primes on intervals $(n, 2n)$ $n\in\mathbb N$, and prove a generalization of it. Extending Erdős' arithmetical-combinatorial argument, we show that for all $k\in\mathbb N$, there is $n_k\in\mathbb N$ such that the intervals $(kn, (k+1)n)$ contain a prime for all $n\geq n_k$. A quantitative lower bound is derived for the number of primes on such intervals. We also give numerical upper bounds for $n_k$ for $k \leq 20$, and we draw comparisons with existing results in the literature.