Prime values of digital functions along the primes

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This event attributed to Claude Opus 5

This event attributed to Claude Fable 5

VibeMathed record: Prime values of digital functions along the primes evidence for this event

Prime values of digital functions along the primes evidence for this event

Prime values of digital functions along the primes parent of this event

Lean 4 formalisation, axiom audit and verification report evidence for this event

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. parent of this event