Jensen Kohlmeyer

math.NT — Number Theory

Spectral synthesis for the Báez–Duarte step space

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer, Liam Kruer

We prove that the closed Báez–Duarte subspace of the Mellin step space is exactly the common kernel of the derivative evaluations prescribed by the zeros of the Riemann zeta function in $\operatorname{Re}s>1/2$, with their multiplicities. No finiteness or sparsity assumption on this zero set is required. The proof combines a uniform fractional regularity estimate for Mellin transforms of step functions with a comparison obtained by reflecting zeros in a shifted vertical line. These estimates produce a holomorphic Hardy-space family whose membership in the Báez–Duarte subspace continues from large positive shifts to every sufficiently small positive shift. A uniformly bounded weak limit then gives the synthesis theorem.

cs.FL — Formal Languages and Automata Theory

Decomposable disjoint-union circuits for primitive words

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer, Liam Kruer

For every fixed finite alphabet $A$ with at least two letters, we construct smooth decomposable circuits representing the primitive words of each length $n$. The gates are literals, Cartesian products on disjoint position sets, and disjoint unions on equal position sets. The total number of gates and wires is $O_A(n^2 5^{\omega(n)})=n^{2+o(1)}$, and the construction takes $n^{2+o(1)}$ bit operations. Prime periods are expressed as coordinate directions of independent rectangular grids. A recursion at the first nonconstant vertex computes the exact set of failed directions; sharing its continuations gives the size bound. This resolves the primitive-word instance of the circuit question posed by Amarilli, Monet, Raphael, and Salvati. It does not determine whether primitive words form a context-free or a poly-slicewise-unambiguous language. The same circuit bound holds for any union of exact power orders at a fixed length.

Author comments: First version. Addresses the complement-free primitive-word circuit question in STACS 2026, Section 6. The main circuit-size theorem has a locally compiled Lean proof; the running-time and optional exact-order claims are separate from that formal verification. Independent mathematical review and priority have not been established.