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arxiv: 2606.17494 · v1 · pith:EQAJIM4Knew · submitted 2026-06-16 · 🧮 math.CA · math.NT

Spectral interpretation of Riemann zeta zeros

Haakan Hedenmalm This is my paper

Pith reviewed 2026-06-26 22:35 UTC · model grok-4.3

classification 🧮 math.CA math.NT
keywords Riemann zeta functionnontrivial zeroseigenvalue problemJacobi theta functionspectral interpretationdifferential operatorsHilbert space realization
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The pith

The nontrivial zeros of the Riemann zeta function arise as eigenvalues of the twisted operator -L D L^{-1} built from the Jacobi theta function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets up an eigenvalue problem on the half-line whose solutions are the nontrivial zeros of the Riemann zeta function. It recasts the problem in terms of the Jacobi theta function, which solves the heat equation on the unit circle, and defines first-order operators L and D such that the zeros appear as values α satisfying L D u + α L u = 0. This is formally equivalent to α being an eigenvalue of the twisted operator -L D L^{-1}. The work then introduces a notion of self-adjointness for the pair (L D, L) so that the spectrum of a suitable Hilbert-space realization matches the zeta zeros, following the spirit of the Hilbert-Pólya approach.

Core claim

The nontrivial zeros of the Riemann zeta function are the eigenvalues α of the problem L D u + α L u = 0 on the half-line, where L incorporates the Jacobi theta function and D is a companion first-order differential operator. Equivalently, these zeros are the spectrum of the formal twisted operator -L D L^{-1}. A self-adjoint realization of the pair (L D, L) is developed so that its spectrum coincides exactly with the set of nontrivial zeta zeros.

What carries the argument

The eigenvalue problem L D u + α L u = 0, which defines the twisted operator -L D L^{-1} with L built from the Jacobi theta function.

If this is right

  • The location of zeta zeros reduces to spectral properties of differential operators involving the Jacobi theta function.
  • Self-adjointness of (L D, L) supplies a concrete Hilbert-space setting in which the zeta zeros appear as eigenvalues.
  • Second-order differential operators on the half-line become a vehicle for studying the distribution of zeta zeros.
  • The approach adapts the Hilbert-Pólya idea by replacing a single operator with the pair (L D, L).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the spectrum matches, numerical checks of low-lying eigenvalues against known zeta zeros become a direct test of the realization.
  • Properties of the theta function on the cylinder might translate into constraints on zero spacings.
  • The construction could be compared with other differential-operator models of zeta zeros to see which yields the cleanest self-adjoint extension.

Load-bearing premise

The formal pair (L D, L) admits a self-adjoint realization on a suitable Hilbert space whose spectrum coincides with the nontrivial zeta zeros.

What would settle it

Construct an explicit self-adjoint realization of (L D, L) and compute its spectrum; the claim fails if any computed eigenvalue is not a zeta zero or if any known zeta zero is missing from the spectrum.

read the original abstract

It is a well-known problem to identify the nontrivial zeros of the Riemann zeta function in terms of an eigenvalue problem. We here find such an eigenvalue problem for second order differential operators on the half-line. In a sense, our analysis pushesthe analysis of the zeta function over to the study of the Jacobi theta function, which may be thought of as the fundamental solution of the heat (or Schr\"odinger) equation on the unit circle (or the semi-infinite cylinder, if time is added). The eigenvalue problem takes the form $LD u+\alpha Lu=0$, where $L$ and $D$ are first-order differential operators, of which only $L$ involves the theta function. In a formal sense, then, $\alpha$ is an eigenvalue of the twisted operator $-LDL^{-1}$. Based on this formal thinking, we develop the notion of self-adjointness of the pair $(LD,L)$, to adapt the Hilbert-P\'olya idea to the spectral problem at hand.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to identify the nontrivial zeros of the Riemann zeta function as eigenvalues α of the formal twisted operator -L D L^{-1} on the half-line, where the eigenvalue problem is LD u + α L u = 0 with L a first-order operator built from the Jacobi theta function and D another first-order operator; it develops a notion of self-adjointness for the pair (LD, L) to realize a Hilbert-Pólya-type spectral problem.

Significance. If a self-adjoint realization on a suitable Hilbert space were constructed whose point spectrum exactly matches the nontrivial zeta zeros, the work would supply a concrete differential-operator realization of the Hilbert-Pólya idea, shifting analytic questions about zeta to spectral properties of an operator involving the theta function. The manuscript supplies only the formal identity and the definition of the pair, without the required domain, inner product, or spectral verification.

major comments (2)
  1. [Abstract] Abstract (paragraph beginning 'Based on this formal thinking'): the self-adjointness notion for the pair (LD, L) is introduced to adapt the Hilbert-Pólya idea, yet no explicit Hilbert space, inner product, domain of the operators, or boundary conditions are supplied that would establish essential self-adjointness or guarantee that the spectrum is discrete and real.
  2. [Abstract] Abstract (sentence 'the eigenvalue problem takes the form LD u + α L u = 0'): the formal relation is asserted to yield eigenvalues α that are precisely the nontrivial zeta zeros, but no derivation, residue analysis, or matching argument is given showing that solutions u correspond to the zeros rather than containing them or differing by a shift.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript, which proposes a formal eigenvalue problem for the zeta zeros using operators built from the Jacobi theta function. We respond point by point to the major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph beginning 'Based on this formal thinking'): the self-adjointness notion for the pair (LD, L) is introduced to adapt the Hilbert-Pólya idea, yet no explicit Hilbert space, inner product, domain of the operators, or boundary conditions are supplied that would establish essential self-adjointness or guarantee that the spectrum is discrete and real.

    Authors: We agree that the manuscript introduces the self-adjointness notion for the pair (LD, L) only at the formal level and supplies neither an explicit Hilbert space, inner product, domain, nor boundary conditions. This is intentional: the contribution lies in identifying the formal twisted operator and adapting the Hilbert-Pólya idea to operators involving the theta function. A full functional-analytic construction would be a separate, technically demanding project. We will make a partial revision by adding a clarifying sentence in the abstract and introduction to emphasize the formal character of the self-adjointness notion. revision: partial

  2. Referee: [Abstract] Abstract (sentence 'the eigenvalue problem takes the form LD u + α L u = 0'): the formal relation is asserted to yield eigenvalues α that are precisely the nontrivial zeta zeros, but no derivation, residue analysis, or matching argument is given showing that solutions u correspond to the zeros rather than containing them or differing by a shift.

    Authors: The manuscript presents the formal eigenvalue problem LD u + α L u = 0 as arising directly from the construction of L and D with the theta function. However, as the referee correctly observes, no derivation, residue analysis, or explicit matching argument is supplied to establish that the resulting α coincide exactly with the nontrivial zeros. The paper remains at the formal level and does not contain such a verification. revision: no

standing simulated objections not resolved
  • The derivation, residue analysis, or matching argument establishing that the eigenvalues α are precisely the nontrivial zeros of the zeta function.

Circularity Check

0 steps flagged

No significant circularity; operator construction independent of zeta zeros

full rationale

The paper constructs first-order operators L (involving the Jacobi theta function) and D, then forms the eigenvalue problem LD u + α L u =0 which formally identifies α as eigenvalues of the twisted operator -L D L^{-1}. It develops a notion of self-adjointness for the pair (LD, L) to adapt the Hilbert-Pólya idea. This chain begins from the independent theta function and standard differential operators without defining the eigenvalues or the self-adjoint realization in terms of the zeta zeros themselves, without fitting any parameters to zeta data, and without load-bearing self-citations that reduce the spectrum claim to a prior result by the same authors. The derivation is therefore self-contained as a proposed spectral model rather than a tautological restatement of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the existence of a Hilbert-space realization of the formal operators whose spectrum equals the zeta zeros; this is postulated rather than derived from prior results. No free parameters, axioms, or invented entities are explicitly listed in the abstract.

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