Regularity of Dedekind Zeta Functions in Zp-Extensions

Definitions and mathematical statement

Let $p$ be a rational prime, let $K$ be a number field, and let $K_\infty/K$ be a ${\mathbb{Z}}_p$-extension: an infinite Galois extension with topological Galois group isomorphic to the additive group ${\mathbb{Z}}_p$ of $p$-adic integers. For each integer $n\geq 0$, let $K_n$ be its unique intermediate field with $[K_n:K]=p^n$. Thus $K_0=K$ and $K_\infty=\bigcup_{n\geq 0}K_n$.

For a number field $F$, write $\mathcal O_F$ for its ring of integers. Its Dedekind zeta function is initially defined by \[ \zeta_F(s)=\sum_{0\neq\mathfrak a\subseteq\mathcal O_F}(\mathrm N\mathfrak a)^{-s},\qquad \operatorname{Re}(s)>1, \] where the sum is over nonzero integral ideals and $\mathrm N\mathfrak a=|\mathcal O_F/\mathfrak a|$, and is subsequently understood by meromorphic continuation. One invariant mentioned in the supplied background is the Euler--Kronecker constant \[ \gamma_F=\lim_{s\to 1}\left(\frac{\zeta_F'(s)}{\zeta_F(s)}+\frac{1}{s-1}\right). \]

Research question. Identify and prove further regularity phenomena for the sequence of Dedekind zeta functions $(\zeta_{K_n})_{n\geq 0}$ and their invariants as $n\to\infty$, beyond the known Iwasawa-type asymptotics referenced by the supplied record. The cleaned record asks for this investigation along arbitrary ${\mathbb{Z}}_p$-extensions, without imposing further restrictions on $p$, $K$, or the extension.

Unresolved formulation. The original question is only "Are there other regularities in ${\mathbb{Z}}_p$-extensions?" It specifies neither a regularity property nor an invariant, normalization, asymptotic scale, or mode of convergence. The supplied reference to Iwasawa's three-parameter asymptotic gives no formula or precise baseline to be excluded. Euler--Kronecker constants are mentioned in the background, but no conjecture concerning them is stated. Consequently, the record determines a research direction rather than a proposition with a specified conclusion; it also does not specify which conclusions should hold for every extension or what uniformity is intended.

Short English statement

Investigate additional patterns in Dedekind zeta functions and their invariants along the finite layers of arbitrary ${\mathbb{Z}}_p$-extensions. The supplied question leaves the proposed patterns unspecified, so a precise conjecture still requires formulation and human mathematical review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30006431$, source problem code catalogue entry, title Regularity of Dedekind Zeta Functions in Zp-Extensions. The supplied record labels the problem open and its statement corrected_verified; these are supplied registry labels, not independent verification.
• Supplied report citation: Cohomology Theories for Automorphic Forms and Enumerative Algebra (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/38, pp. 2065--2124. https://doi.org/10.4171/owr/2025/38. The original excerpt identifies p. 2095 and the contribution Analytic properties of the Dedekind zeta function over Zp-extensions, by Dohyeong Kim, joint work with Harin Jung. No separate proposer is specified.
• Supplied literature source, "Oberwolfach Report 38": https://ems.press/content/serial-article-files/52251.
• Supplied literature source, "Primary report PDF": https://publications.mfo.de/bitstream/handle/mfo/4379/OWR_2025_38.pdf?sequence=1.
• Literature assessment dated 2026-08-22: the supplied curation describes Iwasawa's three-parameter asymptotic as established, mentions current work on further regularity of Dedekind zeta functions and Euler--Kronecker constants, and labels the broad question for arbitrary ${\mathbb{Z}}_p$-extensions open. This is supplied dated curation, not an independently verified literature assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.