Analytic Continuation of Topological Zeta Functions

Definitions and mathematical statement

Let $X$ be a manifold. The supplied record refers to values \[ \zeta_X(m_n) \] of a proposed topological zeta function, described as being defined from limiting homological densities. It supplies neither a definition of these densities nor a formula relating them to $\zeta_X(m_n)$. The index range and the parameters $m_n$ are also unspecified.

The intended question is: for such a manifold $X$, does the proposed function $\zeta_X(s)$ admit analytic continuation to noninteger values of $s$?

Under the usual complex-analytic interpretation, an analytic continuation requires an initially defined holomorphic function $f_X\colon U\to{\mathbb{C}}$ on an open domain $U\subseteq{\mathbb{C}}$, together with a specified continuation domain. For continuation to a larger domain $V\supsetneq U$, the required extension would be a holomorphic function $F_X\colon V\to{\mathbb{C}}$ satisfying \[ F_X|_U=f_X. \] The supplied record does not specify such an initial function or either domain. Prescribing values at parameters $m_n$ alone does not specify an initial holomorphic function; asking for a holomorphic function agreeing with those values would instead be an interpolation question unless further structure is supplied.

Consequently, the displayed question records the intended target but is not yet a fully specified analytic-continuation problem. The manifold hypotheses, density construction, parameter sequence, initial analytic definition, and intended continuation domain must be recovered before a precise statement can be completed.

Short English statement

The question asks whether a proposed zeta function obtained from limiting homological densities of a manifold extends analytically to noninteger arguments. The supplied record omits the density formula, parameter sequence, manifold hypotheses, and initial analytic domain, so the intended continuation problem remains underspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local problem-registry record, permanent UnsolvedMath ID $30003181$, title "Analytic Continuation of Topological Zeta Functions," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2016.
• Supplied report citation: Topologie (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/35, pp. 2009--2068. https://doi.org/10.4171/owr/2016/35. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. The citation's 2017 label and the record's 2016 proposed year are retained without reconciliation.
• Farb--Wolfson--Wood, "homological densities" (supplied reference label). https://doi.org/10.1016/j.aim.2019.06.016.
• Farb--Wolfson--Wood, preprint (supplied reference label). https://arxiv.org/abs/1611.04563.
• Supplied dated literature curation, checked 2026-08-21: the record labels the problem open and assesses that the homological-density identities were proved, while analytic continuation of the proposed topological zeta function has not been established. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The supplied curation expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and states that its mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. This is a supplied editorial qualification; it does not resolve the missing definitions in the present record or constitute mathematical verification of this draft.