Discoveries / number-theory

Prime values of digital functions along the primes

For every integer-valued strongly $b$-additive $g$ with $\gcd(g(1),\dots,g(b-1))=1$ and digit mean $\mu_g\ge0$: $g(p)$ is prime for infinitely many primes $p$. For $\mu_g>0$, $\sum 1/p$ over $p<X$ with $g(p)$ prime is $(d_g/\varphi(d_g))\log_3X + C_{g,1} + O(1/\log\log X)$, likewise for the first $j$ iterates. Also $\#\{p\le x: g(p)\text{ prime}\}\ll\pi(x)/\log\log x$, of that exact order on a large set of $x$, and $\omega(g(p))$ has normal order $\log_3 p$. What is new and what is not. For the digit sum $g=s_b$, Harman (2012) already proved both the infinitude and a Mertens formula; the new information there is the remainder tending to a limit rather than being $O(1)$, and the iterated version for $g=s$ is, in the paper's words, "contained, in a stronger and quantitative form, in Harman". The new content is the generalization to every such $g$, which delivers the running example $g=S$, the sum of squared decimal digits, and so the infinitude of OEIS A052034.