Pseudodifferential arithmetic, Riemann and Lindel\"of hypotheses
Pith reviewed 2026-05-24 10:58 UTC · model grok-4.3
The pith
The Riemann hypothesis is equivalent to estimates on a Weyl-calculus operator from zeta zeros, and explicit construction via pseudodifferential arithmetic disproves a measure conjecture on zero real parts while proving the Lindelöf
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The Riemann hypothesis is equivalent to the validity of a collection of estimates involving an operator constructed from a distribution decomposing over the zeros of the Riemann zeta function using the Weyl symbolic calculus. Pseudodifferential arithmetic makes the operator fully explicit, leading to a disproof of the conjecture that the closure of the set of real parts of non-trivial zeros of zeta has measure at least 0.5, and to a proof of the Lindelöf hypothesis.
What carries the argument
The operator from the Weyl symbolic calculus with symbol a distribution over zeta zeros, made explicit using pseudodifferential arithmetic.
If this is right
- The estimates on the operator are equivalent to the Riemann hypothesis.
- The closure of real parts of non-trivial zeta zeros has measure strictly less than 0.5.
- The Lindelöf hypothesis holds, bounding the growth of the zeta function on the critical line.
Where Pith is reading between the lines
- The explicit operator may permit numerical or symbolic checks of the estimates without prior assumptions on zero locations.
- Similar explicit constructions could extend to other L-functions beyond the zeta function.
- The narrowed measure on real parts constrains possible clustering of zeros more tightly than the disproved conjecture allowed.
Load-bearing premise
The pseudodifferential arithmetic yields an operator form that permits direct verification of the estimates without any hidden circular reliance on the locations of the zeros themselves.
What would settle it
An independent calculation of the measure of the closure of the real parts of the non-trivial zeros of the zeta function; a result of 0.5 or more would refute the disproof.
read the original abstract
The Weyl symbolic calculus of operators leads to the construction, if one takes for symbol a certain distribution decomposing over the zeros of the Riemann zeta function, of an operator with the following property: the Riemann hypothesis is equivalent to the validity of a collection of estimates involving this operator. Pseudodifferential arithmetic, a novel chapter of pseudodifferential operator theory, makes it possible to make the operator under study fully explicit. This leads to a disproof of the conjecture: the closure of the set of real parts of non-trivial zeros of zeta is dense in $(0,1)$. A similar method leads to a proof of the Lindel\\"of hypothesis.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript constructs an operator via the Weyl symbolic calculus whose symbol is a distribution that decomposes over the non-trivial zeros of the Riemann zeta function. It asserts that the Riemann hypothesis is equivalent to a collection of estimates on this operator. A new framework called pseudodifferential arithmetic is introduced to render the operator fully explicit, from which the author derives a disproof of the conjecture that the closure of the set of real parts of the non-trivial zeros has Lebesgue measure at least 0.5, together with a proof of the Lindelöf hypothesis.
Significance. If the central derivations are free of circular dependence on the zero distribution and the explicit operator yields verifiable estimates, the results would resolve two major open problems in analytic number theory and introduce a potentially powerful new calculus. The approach is highly original, but its significance is conditional on rigorous demonstration that the arithmetic steps do not presuppose the very distribution being constrained.
major comments (3)
- [Abstract] Abstract: the symbol is introduced as a distribution decomposing over the non-trivial zeros. The subsequent claim that pseudodifferential arithmetic produces an explicit operator whose norm or trace estimates are independent of zero locations must be accompanied by an explicit verification (e.g., a step-by-step expansion of the symbol-to-operator map) showing that no identity or bound invokes the density or positions of those zeros; absent this, the asserted equivalence to RH and the disproof of the measure-≥0.5 conjecture reduce to tautologies.
- [Abstract] Abstract (equivalence claim): the statement that RH is equivalent to 'a collection of estimates' on the operator is asserted without derivation or explicit formulas. The manuscript must supply the precise estimates (norm bounds, trace formulas, or spectral conditions) together with a proof that they are non-circular and that their validity is strictly equivalent to the non-vanishing of zeta on Re(s)=1/2.
- [Abstract] Abstract (Lindelöf proof): the claim that a similar method proves the Lindelöf hypothesis likewise requires the explicit operator and the precise growth estimate derived from it; without an error-controlled derivation that avoids feeding the zero distribution back into the bounds, the proof cannot be accepted as independent of the hypothesis under study.
minor comments (1)
- The abstract refers to 'pseudodifferential arithmetic' as a novel chapter without a self-contained definition or reference to its foundational axioms; a brief outline of its key rules (e.g., composition or symbol calculus identities) would improve accessibility.
Simulated Author's Rebuttal
We thank the referee for the careful reading and for identifying points where greater explicitness would strengthen the presentation. The manuscript develops the operator via pseudodifferential arithmetic precisely to avoid circular dependence on zero locations; the requested verifications appear in the body but will be highlighted and expanded for clarity.
read point-by-point responses
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Referee: [Abstract] Abstract: the symbol is introduced as a distribution decomposing over the non-trivial zeros. The subsequent claim that pseudodifferential arithmetic produces an explicit operator whose norm or trace estimates are independent of zero locations must be accompanied by an explicit verification (e.g., a step-by-step expansion of the symbol-to-operator map) showing that no identity or bound invokes the density or positions of those zeros.
Authors: The full text constructs the operator through a sequence of formal operations in the Weyl calculus and the newly introduced pseudodifferential arithmetic; these steps operate on the distributional symbol using only its algebraic and continuity properties, without inserting any information about zero density or positions. To make the independence fully transparent, we will add a dedicated appendix containing the requested step-by-step expansion of the symbol-to-operator map together with a verification that no bound relies on zero locations. revision: yes
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Referee: [Abstract] Abstract (equivalence claim): the statement that RH is equivalent to 'a collection of estimates' on the operator is asserted without derivation or explicit formulas. The manuscript must supply the precise estimates (norm bounds, trace formulas, or spectral conditions) together with a proof that they are non-circular and that their validity is strictly equivalent to the non-vanishing of zeta on Re(s)=1/2.
Authors: Section 4 of the manuscript derives the equivalence by showing that the operator norm bounds in appropriate Sobolev spaces are equivalent, via the explicit form obtained from pseudodifferential arithmetic, to the absence of zeros off the critical line. We will extract the precise norm bounds and the equivalence proof into a new self-contained subsection so that the non-circular character is immediately visible. revision: yes
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Referee: [Abstract] Abstract (Lindelöf proof): the claim that a similar method proves the Lindelöf hypothesis likewise requires the explicit operator and the precise growth estimate derived from it; without an error-controlled derivation that avoids feeding the zero distribution back into the bounds, the proof cannot be accepted as independent of the hypothesis under study.
Authors: The Lindelöf proof proceeds from the same explicit operator by deriving a growth bound on its symbol that translates directly into the required zeta growth on the critical line; the derivation uses only the arithmetic rules and does not presuppose any zero distribution. We will insert the detailed, error-controlled growth estimate as a separate proposition with a clear statement of the independence from the hypothesis. revision: yes
Circularity Check
Symbol defined by decomposition over zeta zeros makes subsequent RH-equivalence and Lindelöf claims reduce to the input distribution
specific steps
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self definitional
[Abstract]
"if one takes for symbol a certain distribution decomposing over the zeros of the Riemann zeta function, of an operator with the following property: the Riemann hypothesis is equivalent to the validity of a collection of estimates involving this operator. Pseudodifferential arithmetic... makes it possible to make the operator under study fully explicit. This leads to a disproof of the conjecture: the closure of the set of real parts of non-trivial zeros of zeta has measure at least 0.5. A similar method leads to a proof of the Lindelöf hypothesis."
The symbol is explicitly constructed as a distribution that decomposes over the zeros; the operator is then made explicit from this symbol and its estimates are declared equivalent to RH (and sufficient to disprove the measure conjecture and prove Lindelöf). The estimates therefore cannot be independent of the zero locations used to define the symbol; the claimed implications reduce to properties already built into the input distribution.
full rationale
The paper constructs its central operator by taking a symbol that decomposes over the non-trivial zeros; the claimed equivalence between operator estimates and RH, plus the disproof of the measure-0.5 conjecture and the Lindelöf proof, are then asserted to follow from pseudodifferential arithmetic applied to that operator. Because the symbol is built directly from the zeros whose distribution is the object of study, any estimate derived from the operator necessarily encodes information about those same zeros, rendering the logical chain self-definitional rather than an independent derivation. No external benchmark or machine-checked identity is cited to break the dependence. This matches the self-definitional pattern at the level of the symbol definition itself.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math The Weyl symbolic calculus applies to the chosen distribution symbol built from zeta zeros.
- ad hoc to paper Pseudodifferential arithmetic renders the operator fully explicit.
invented entities (1)
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The operator constructed from the zeta-zero distribution symbol
no independent evidence
discussion (0)
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