Extended Riemann Hypothesis (ERH) for Dedekind Zeta Functions

[franzhusch]

What is the conjecture

The Extended Riemann Hypothesis (ERH) concerns zeros of Dedekind zeta functions of algebraic number fields. Let $K$ be an algebraic number field and let $\zeta_K(s)$ denote its Dedekind zeta function. The Extended Riemann Hypothesis asserts that every non-trivial zero of $\zeta_K(s)$ that lies in the critical strip ${s \in \mathbb{C} : 0 < \text{Re}(s) < 1}$ has real part equal to $\frac{1}{2}$. Equivalently, all such zeros lie on the critical line $\text{Re}(s) = \frac{1}{2}$. The ERH is a natural generalization of the classical Riemann Hypothesis from the Riemann zeta function to the broader class of Dedekind zeta functions.

(This description may contain subtle errors especially on more complex problems; for exact details, refer to the sources.)

Sources:

• https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis, https://arxiv.org/abs/2509.21518, https://encyclopediaofmath.org/wiki/Riemann_hypothesis,_generalized

Prerequisites needed

Formalizability Rating: 3/5 (0 is best) (as of 2026-02-21)

Building blocks (1-3; from search results):

• Algebraic number field structures (available in Mathlib)
• Basic complex analysis and analytic continuation (available in Mathlib)
• Special function definitions (some support in Mathlib)

Missing pieces (exactly 2; unclear/absent from search results):

• Dedekind zeta function definition and its analytic properties for general number fields
• Characterization of the critical strip and zeros of analytic functions in the required form

Rating justification (1-2 sentences): Formalizing the ERH statement requires developing the full theory of Dedekind zeta functions (definition, meromorphic continuation, functional equation) for arbitrary number fields, which is substantial new infrastructure not present in Mathlib. However, once these definitions exist, the conjecture statement itself can be formulated relatively directly.

AMS categories

• ams-11
• ams-12
• ams-30

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