When Euler met Brun (Q6595579)

Let \(p'\) denotes the smallest prime that is larger than prime \(p\). Brun's sieve asserts that for any fixed \(K\), \(\sum_{p'-p \le K}1/p<\infty\). In the paper under review, motivated by replacing \(K\) by a growing function, say \(y(p)=\lambda (p) \log p\), the author surveys some results concerning the following sum\N\[\NS(x):=\sum_{\substack{p\le x\\ p'-p \le y(p)}}\frac{1}{p}.\N\]\NIf for \(k\ge 2\) we let \(\log_k\) to be the \(k\)-th iterated logarithm, and define the ``logorial'' function by \(\mathrm{Log}_k = \prod_{2\le j \le k}\log_j\), then the author shows that as \(x \to \infty\)\N\[\NS(x)\sim \log_{k+1}x\N\]\Nfor \(\lambda (p) = 1/\mathrm{Log}_k(p)\), but \(S(x)\ll 1\) for \(\lambda (p) = 1/\mathrm{Log}_k(p)(\log_k p)^{\varepsilon}\), under the assumption that\N\[\N\frac{1}{\pi(x)}\#\left\{p\le x; \frac{p'-p}{\log p} \le \lambda(p)\right\} \sim \int^{\lambda(x)}_{0}e^{-u}du = 1-e^{-\lambda(x)}\N\]\Nholds for these functions \(\lambda\). The author also considers the problem unconditionally using sieve bounds.