Periodic counting, carry synchronization, and primes in nonstandard numeration systems

Abstract

Let $b\ge2$ be an integer and let $D\subset\mathbb{N}_0$ be a finite digit set containing zero, with maximum residue multiplicity $μ=\max_{0\le r<b}\#\{d\in D:d\equiv r\pmod b\}$ and cardinality $q=\#D\ge2$. Under the condition $μ<q$, we prove that the summatory representation function satisfies an asymptotic formula with a strictly positive, Hölder continuous, $1$-periodic fluctuating profile and an error of $O_D(x^{\log_bμ})$. An exact periodic factorization of the Laplace generating function furnishes an explicit, computable Fourier expansion for the periodic profile via Gamma factors, and yields the meromorphic continuation of the associated Dirichlet series to the entire complex plane. When $D$ is a complete residue system modulo $b$, we establish an arithmetic synchronization criterion showing that the associated finite carry system admits a zero-synchronizing word if and only if $\gcd(D)=1$. This synchronization yields an exponentially accurate interval approximation and a general transfer principle for arithmetic weights. Consequently, we derive a prime number theorem for representable primes governed by the same periodic profile, alongside an exponentially convergent measurable scaling limit of the representable sets.