A Bombieri-Vinogradov-type theorem for moduli with small radical

In this article, we extend our recent work on a Bombieri-Vinogradov-type theorem for sparse sets of prime powers $p^N\le x^{1\mathrm{/}4-\epsilon }$ with $p\le (\log x{)}^C$ to sparse sets of moduli $s\le x^{1\mathrm{/}3-\epsilon }$ with radical rad$(s)\le x^{9\mathrm{/}40}$. To derive our result, we combine our previous method with a Bombieri-Vinogradov-type theorem for general moduli $s\le x^{9\mathrm{/}40}$ obtained by Roger Baker.