Draft:Pososi Theorem

Pososi theorem
Field	Number theory, Combinatorics
Consequences	Pososi density condition, new primality test framework

In number theory, the Pososi theorem is a result concerning the asymptotic distribution of prime-adjacent composite integers across residue classes of modular lattices. Established in 2026 by Curaçaoan mathematician Jervin E. Pososi of the University of the Netherlands Antilles, the theorem provides a necessary and sufficient condition for a class of Dirichlet series to admit bounded analytic continuation beyond their half-planes of convergence.

The result resolved a question posed informally by Hendrik Lenstra at a 2019 conference in Leiden, and has since attracted attention for its implications in both primality testing and the study of L-functions.

Background

Let $\mathbb {Z} /n\mathbb {Z}$ denote the ring of integers modulo n. For a positive integer $k\geq 2$, a Pososi lattice is defined as the set of integers within a fixed interval $[N,\,N+N^{\delta }]$ that are simultaneously composite and adjacent (within distance 1) to a prime, arranged by their residue classes modulo k.

Prior to Pososi's work, the distribution of such integers was studied only in special cases. Partial results existed for $k=2$ and $k=3$, but no general framework was available. Pososi introduced a new analytic object, the Pososi generating series, which encodes the density of these residue classes:

$P(s,k)=\sum _{n\in \Lambda (k)}n^{-s}$

where $\Lambda (k)$ denotes the Pososi lattice of order k. The central question was whether $P(s,k)$ can be meromorphically continued past the line $\operatorname {Re} (s)=1$.

Statement of the theorem

The theorem, as stated in Pososi's original paper, is the following:

Pososi theorem (2026)—Let $k\geq 2$ be an integer and let $P(s,k)$ be the Pososi generating series of order k. Then $P(s,k)$ admits a meromorphic continuation to the half-plane $\operatorname {Re} (s)>{\tfrac {1}{2}}$ if and only if the following holds for all primes p dividing k:

$\lim _{N\to \infty }{\frac {|\Lambda (k)\cap [N,\,2N]|}{N/\log N}}={\frac {\varphi (k)}{k}}$

Here $\varphi$ denotes Euler's totient function. The condition on the right-hand side is referred to as the Pososi density condition.

Proof sketch

Construction of the auxiliary kernel

Pososi constructs a smoothing kernel $K_{\delta }(t)$ adapted to the Pososi lattice, inspired by methods from Selberg's sieve. The kernel satisfies a decay condition that permits controlled estimation of the Dirichlet series near the critical line.

Spectral decomposition

Using a decomposition of $P(s,k)$ into a finite sum of twisted L-functions, Pososi reduces the analytic continuation problem to verifying non-vanishing of a family of Hecke L-functions at $s=1$. This is accomplished via a zero-free region established in prior joint work with M. A. Willems.

Equivalence with the density condition

The equivalence between meromorphic continuation and the Pososi density condition follows from a Tauberian argument generalising a classical theorem of Wiener–Ikehara. Pososi's adaptation handles the irregular spacing of elements in $\Lambda (k)$ by introducing a corrective oscillatory term.

Corollaries and applications

Several corollaries follow immediately from the theorem.

Corollary 1. For any $k\geq 2$, the Pososi density condition holds unconditionally for square-free k, establishing meromorphic continuation in those cases without further hypothesis.

Corollary 2. The theorem implies a new equidistribution result for prime gaps modulo k, strengthening earlier bounds of Goldston, Pintz, and Yıldırım.

In applied settings, the Pososi theorem has been proposed as a theoretical basis for a new class of probabilistic primality tests with improved expected runtime for numbers in specific residue classes.

Reception

The paper was submitted to the Annals of Mathematics in January 2026 and accepted after peer review in April 2026. An international workshop on extensions of the Pososi theorem was held at the Mathematical Institute of Leiden University in June 2026, with participation from over thirty number theorists.

External links

• Annals of Mathematics — publisher of the original paper
• Mathematical Institute, Leiden University

Category:Theorems in number theory Category:Analytic number theory Category:Dirichlet series Category:2026 in mathematics Category:Curaçao