An Explicit Surjectivity Threshold for Digit Sums of Primes

Abstract

Let $s(n)=s_{10}(n)$ be the decimal sum-of-digits map. Building on the circle-method framework of Drmota-Mauduit-Rivat for digital restrictions on primes, we make the constants explicit at the points needed to obtain an effective surjectivity statement for digit sums of primes. We exhibit an explicit integer $M < 1.78 \times 10^{32}$ such that every integer $m \ge M$ with gcd$(m,9)=1$ occurs as $s(p)$ for at least one prime $p$. We also prove an explicit lower bound $A_m(10^{2m/9}) \ge C_q(m) 10^{2m/9}/m^{3/2}$, where $C_q(m)$ is explicit, positive above the sufficient threshold, and bounded away from $0$ along each admissible residue class. Existence of a non-numerical threshold follows from the DMR asymptotic theory and was noted by Harman; to the best of our knowledge, this is the first published explicit numerical threshold for this surjectivity statement. The proof combines explicit major-arc estimates, a fully explicit replacement for DMR's implicit prime exponential-sum input, and constant-tracked Type-II minor-arc estimates. As an application, we prove the infinitude of OEIS A070027, the primes whose iterated digit-sum chain remains prime until reaching a one-digit prime. We also record related effective consequences for additive primes and digit-sum additive decompositions.