Forbidden conductors and sequences of $\pm 1$s

TL;DR

The paper proves that forbidden conductors for degree-2 L-functions in the extended Selberg class are dense in the interval $(0,4)$ and that forbidden rational conductors have positive-accumulation points, notably near $\frac{3\pm\sqrt{5}}{2}$. It develops a detailed combinatorial framework around loops $L(q)$ defined via $c(q,\mathbf m)$ and weights $w(q,\mathbf m)$, derives a necessary structure for nontrivial loops when $2<q<4$, and constructs loop families that force non-unit weights to establish density via a Darboux-type argument. The second main result shows accumulation points of rational forbidden $q$ by analyzing the loop $(1,-1,1,-1,c)$ and connecting the required norm conditions to units in the quadratic field $\mathbb{Q}(\sqrt{5})$, enabling approximation of the critical algebraic numbers by rationals. Overall, the work resolves a posed problem on density and reveals the arithmetic nature of forbidden conductors through continued-fraction machinery and unit equations in number fields.

Abstract

We study "forbidden" conductors, i.e. numbers q > 0 satisfying algebraic criteria introduced by J. Kaczorowski, A. Perelli and M. Radziejewski [Acta Arith. 210 (2023), 1-21], that cannot be conductors of L-functions of degree 2 from the extended Selberg class. We show that the set of forbidden q is dense in the interval (0,4), solving a problem posed in [Acta Arith. 210 (2023), 1-21]. We also find positive points of accumulation of rational forbidden q.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Lemma 4
• proof
• Lemma 5
• proof
• Proposition 6
• proof
• Lemma 7
• proof