Nonvanishing of the p-adic zeta function at even integers

Definitions and mathematical statement

Let $p$ be the intended prime, let $\mathbb Z_p$ be the ring of $p$-adic integers, and let $\mathbb Q_p$ be its fraction field. The source denotes a $p$-adic zeta function by $\zeta_p$ and asks \[ \zeta_p(k)\ne0\qquad\text{for every even integer }k. \] Literally the quantifier is $k\in2\mathbb Z$, including zero and negative even integers; a positive-even-only version would need to be stated separately.

The function is not defined by an interpolation formula or branch convention in the supplied record. Standard $p$-adic zeta/$L$-function constructions distinguish character branches and Euler-factor normalizations, and a usual trivial-character zeta function is meromorphic with a pole at $1$, rather than a finite-valued map on all of $\mathbb Z_p$. The even-integer evaluations avoid $1$, but the declared domain still needs correction or explanation. No particular branch is silently selected, since its special values and trivial-zero behavior can affect the nonvanishing claim.

With the intended function and argument range fixed, the target is pointwise nonvanishing at every such integer for each applicable prime, not absence of all $p$-adic zeros on its full domain. The supplied summary links the issue to Vandiver/Leopoldt/Iwasawa questions but gives no exact equivalence or hypotheses.

Short English statement

Is the intended $p$-adic zeta function nonzero at every even integer? Fix its branch, interpolation convention, pole treatment, and whether negative and zero arguments are included.

Sources

• Christian Wuthrich, Open problems in Number Theory (2011), distinct open statement 7, https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf; accessed 2026-07-29, PDF-text/visual-deduplication provenance.
• The supplied summary mentions Vandiver's conjecture, Leopoldt-type conjectures, and Iwasawa-theoretic zero structure without exact theorem statements or separate publication citations.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $8900007$, source code catalogue entry, Nonvanishing of the p-adic zeta function at even integers. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.