A quantitative two-interval criterion for the Riemann hypothesis

Payam Danesh

REVIEW

A quantitative two-interval criterion for the Riemann hypothesis

Payam Danesh

Number Theory · Submitted 7 Nov 2026 · 8 references

Pith's verdict: A clean new equivalence for RH with subexponential supports, logically coherent but sketchy on a few estimates.

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arXiv ID2611.10726 (version 1)

Pith IDCFQVMBZY, Pith's permanent record of this paper and its review

CitationPayam Danesh. "A quantitative two-interval criterion for the Riemann hypothesis." arXiv:2611.10726 (2026). https://pith.science/paper/CFQVMBZY

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Editor's note

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Here's the gist: the paper proves that the Riemann hypothesis is equivalent to the Cauchy–Schwarz inequality R(f,g) ≤ 1 for all admissible pairs (f,g) whose supports lie in two intervals of width w(L) that shrink subexponentially as the separation L grows. If RH fails, the normalized cross term exceeds e^{ηL} for any η < β−1/2. This is a real extension of earlier work by Yoshida and Bombieri, which dealt with fixed or slowly separating supports. The proof combines a smooth partition argument (Lemma 2) with a Laplace transform pole obstruction (Lemma 3) to transfer a bound on admissible pairs to a bound on translates of a fixed test function. The logic is well-structured, and the paper honestly states its limitations: exponential-width decay (w(L) = e^{-cL}) is not covered, and the first violating pair is not constructed. The arithmetic reformulation in §5 is a nice bonus.

What's good: the central idea is original and clearly explained. The estimates in Lemma 2 are sketched but the reasoning is plausible; the partition-of-unity construction and the zero-counting bound are standard, and the constants could be extracted with a bit of work. Lemma 3 is a standard residue argument, correctly applied. The paper does not assume RH or any unproven conjecture, and the forward direction (RH ⇒ R≤1) is just Cauchy–Schwarz, which is trivial but honest.

Soft spots: the proof of Lemma 2 depends on a partition-of-unity whose error terms are not fully traced—bounds like '≪_h log(2+m)/m' are given without explicit constants, and the uniformity in the smoothness of h is assumed rather than checked. The bound for the off-diagonal terms (τ_t u_k, u_j) uses Lemma 1 to guarantee positive denominator, but the transition from admissible-pair hypothesis to the bound on each piece requires that the translated supports land inside the required intervals, which is handled geometrically but the argument could be tightened. These are minor: the paper is not claiming a fully optimized estimate, only the existence of a constant.

Who should read this: number theorists working on explicit formulas and equivalent formulations of RH, especially those interested in Weil's quadratic form. The paper is a serious contribution that deserves peer review.

Recommendation: send to a good number theory journal. The referee will want to check the partition estimate in Lemma 2 carefully and confirm that the subexponential condition is both sufficient and necessary for the equivalence as stated. But the result is likely correct and clearly worth publishing.

Referee report

Significance

If the arguments can be fully verified, this result provides a new, quantitatively explicit equivalence for the Riemann hypothesis. The criterion reduces RH to a uniform inequality on a concrete family of test functions with widely separated, shrinking supports, and it attaches exponential growth rates to the violation when a zero off the line exists. The proof strategy—combining positivity on short intervals, a partition‑of‑unity that keeps the cost subexponential, and a meromorphic continuation obstruction—is clever and draws on classical tools in a nontrivial way. The work thus deepens the connection between Weil positivity and the location of zeta zeros, and the explicit dependence on the width function (including the reciprocal‑width case w(L)=1/L) gives falsifiable predictions that could, in principle, be targeted by numerical experiments.

Major objections (3)

[§3 Lemma 2] The proof of Lemma 2 passes from the hypothesis “R(f,g) ≤ e^{ηL} for every real admissible pair at all sufficiently large L” to a bound on |B(τ_t h, h)|. The constant C_{h,A,η} in (16) is stated to be independent of t, but its dependence on the partition threshold m(t) and on the norm of the decomposition u_j is not made explicit. Since the application in Theorem 4 requires that the same constant works for all t and that the error from the subexponential factor in (17) does not spoil the limit argument, the author should either give a fully explicit estimate or explain why the implied constants can be chosen uniformly. Without this, the transfer from the admissible‑pair bound to the translated bound, and hence the contradiction in the second part of Theorem 4, is not fully quantitative.
[§4 Proof of Theorem 4, second part] The proof of the violation direction uses Lemma 3 with α = η+ε, relying on Lemma 2 to produce a bound of the form O(e^{(η+ε)t}) for the translated function F_h(t). However, Lemma 2 assumes that R(f,g) ≤ e^{ηL} for every real admissible pair at all sufficiently large L, whereas the contrapositive assumption in Theorem 4 is that (30) fails (i.e., R(f,g) ≤ e^{ηL} for all real pairs). This is exactly the hypothesis of Lemma 2, so the implication is logically sound. The only subtlety is the “sufficiently large L” condition: the threshold for which Lemma 2 applies must be independent of the particular h chosen in Lemma 3. The paper should address why the construction of h (which depends on a parameter r) can be fixed to satisfy both the support bound A and the condition that the partition width m(t) yields a valid transfer. This is a uniformity question that, while likely harmless, deserves a brief remark.
[References] The paper cites the author’s own unpublished preprint [7] (“Riemann Zeros as quantum energy levels”, Jul 2026). In the Introduction, [7] is mentioned as a “related spectral formulation” that associates a quadratic energy to each zeta zero. This citation does not appear to be used in any of the lemmas or the main theorem, so it does not affect the logical chain. Nevertheless, because the result is not yet peer‑reviewed and its relevance to the present argument is not essential, the author should either clarify that [7] is not needed for the proof, or, if any part of the current work relies on it, provide a verification or state that reliance. The current framing leaves this ambiguity.

Minor objections (5)

[§2, Eq. (10)] The estimate |ˆf(y)| ≤ (d/(2π))^{1/2} ‖f‖_2 is derived from Cauchy–Schwarz and the support length d. While correct, the factor d/(2π) is the square of an L¹→L∞ bound, and the step “by Plancherel, at least half the squared norm lies outside this interval” relies on the specific choice M = π/(2d) which makes |∫_{-M}^{M} |ˆf|^2 dy| ≤ ½‖f‖_2^2. This is a clean trick, but a brief justification that the pointwise bound indeed gives that inequality (rather than a strict inequality) would improve readability.
[§3, Eq. (19)] The integration variable in (19) runs from −1 to 1 after the substitution v = m x. The exponents contain σ v / m and i y v / m, which are consistent. However, the sentence “The amplitude and its first two derivatives in v are bounded independently of m, j, σ” is vague. Since σ is confined to |σ| ≤ 1/2 and v ∈ [−1,1], the factor e^{σ v / m} is indeed bounded, but the derivatives with respect to v of the integrand involve σ/m factors. A short comment or an explicit bound would remove any doubt about the uniformity claimed.
[§3, Eq. (20)] The bound |H_{τ_{-j/m} u_j}(σ+iy)| ≤ C_h m^{-1} (1+|y|/m)^{-2} is stated without a proof. While the idea (direct bound for |y|≤m, two integrations by parts for |y|>m) is standard, the constant C_h should be made explicit or at least described as depending on finitely many derivatives of h and χ. In particular, the statement “the boundary terms vanish since χ is smooth and compactly supported” is correct, but the decay rate of the derivatives of the transform as a function of y should be checked against the factor e^{σ v / m} when |y| is large. Including a short estimation would strengthen the presentation.
[§1, Figure 2] The caption reads “three examples of these analytic functions” but the plotted functions are w(L)=L^{-1}, L^{-2}, exp(−√L), which are not necessarily analytic at the origin (the exponential power is subanalytic). “Analytic” is misleading; “decreasing width functions” or “examples satisfying (12)” would be more accurate.
[§5, Eq. (35)] The integrand e^{-u/2} / (1−e^{-2u}) is well‑defined for u>0, but the paper does not mention that the integral is absolutely convergent because the support of k(u) stays away from 0 (u ≥ L−2w(L) → ∞). Adding a brief remark would improve clarity.

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The paper

The paper claims

The Riemann hypothesis is equivalent to a Cauchy-Schwarz bound on test functions with subexponentially shrinking supports.

The reading

This paper tries to prove that the Riemann hypothesis (RH) can be reformulated as a concrete inequality involving the Cauchy-Schwarz norm of the Weil quadratic form evaluated on pairs of smooth test functions. If RH is true, then for every pair of functions supported on two short intervals that move far apart and shrink subexponentially, the normalized cross term is bounded by 1. If RH is false (a zero with real part greater than 1/2 exists), then the cross term grows exponentially at arbitrarily large separations, with an exponent that can be made arbitrarily close to the zero's horizontal displacement. The key advance is that the support intervals can be made extremely narrow (width decaying faster than any exponential) while still capturing the full force of RH. The resulting criterion pinpoints a uniform arithmetic inequality involving prime powers in a moving interval whose proof would settle RH.

What carries the argument

The central object is the normalized Weil cross term R(f,g)=|B(f,g)|/√(Q(f)Q(g)), where B is the Weil quadratic form summing over zeta zeros. The machinery that carries the argument is a smooth partition of unity that decomposes a fixed test function h into O(1/w(L)) pieces, each supported on an interval of width about w(L), and then applies the hypothesized bound on admissible pairs to the translates. The subexponential condition log(1/w(L))=o(L) makes the number of pieces subexponential in L, so that any exponential growth from a counterexample survives the summation. The pole obstruction in Lemma 3 uses the Laplace transform of the translated cross term to show that an exponential bound would force holomorphy in a half-plane contradicting a residue from a zero off the critical line.

Extended reading notes

Core claim

The paper establishes that the Riemann hypothesis is equivalent to the condition that for every real admissible pair of smooth functions (f,g) — supported respectively on [0,w(L)] and [L-w(L),L] with log(1/w(L))=o(L) — the normalized Weil cross term R(f,g) ≤ 1 for all sufficiently large L. If RH holds, Cauchy-Schwarz on the zero sequence gives the bound directly. If RH fails, a zero with real part β>1/2 forces the existence of pairs with R(f,g) > $e^{{ηL}}$ for any η < β-1/2, violating the bound. The proof combines a short-support positivity estimate (Lemma 1), a smooth partition procedure that transfers a bound on narrow-supported pairs to translates of a fixed function with a subexponential cost (Lemma 2), and a Laplace-transform pole argument showing that an exponential bound on the translated term would contradict the presence of an off-line zero (Lemma 3). The width condition log(1/w(L))=o(L) is essential: it ensures the partition cost remains subexponential, so the exponential violation from a false zero cannot be absorbed.

Load-bearing premise

The subexponential decay condition log(1/w(L))=o(L) on the support width is load-bearing: it ensures the number of partition pieces needed to translate a bound from narrow to wide functions grows subexponentially, so that an exponential violation from a false zero cannot be hidden by the partition cost.

Editorial extensions

If this is right

If the paper's equivalence is correct, proving RH becomes a single uniform arithmetic inequality: for all admissible pairs with subexponentially shrinking supports, the prime-power sum and the pole-integral combination in equation (36) must not exceed √(Q(f)Q(g)), with a constant gap that is uniform in L.
The reciprocal-width case w(L)=1/L is a concrete test case: one would need to show that for functions supported on intervals of length 1/L, the normalized cross term never exceeds 1.
Any computational or analytic method that verifies the inequality for a class of admissible pairs would immediately imply RH for that class; a single counterexample (R>1) at some large L would disprove RH.
The result shows that the Weil positivity criterion does not require full flexibility in support geometry — two widely separated, extremely narrow intervals suffice, which may simplify future numerical or analytic attacks on RH.
The paper's Lemma 2 gives a template for transferring bounds from narrow-supported pairs to much wider functions, with the subexponential cost controlled entirely by the width decay rate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the equivalence holds, searching for a counterexample to RH could be replaced by searching for a pair (f,g) of the prescribed narrow-support form with R(f,g)>1. Since the supports are extremely short, such a search might be computationally tractable for moderate L, but the required L may be astronomically large before the exponential violation kicks in.
The width condition log(1/w(L))=o(L) includes slowly decaying widths like 1/L^a for any a>0, but excludes exponential decay w(L)=e^{-cL}. This suggests a potential sharpness: if w(L) decays exponentially, the partition cost becomes exponential and the argument fails; determining whether the equivalence still holds for such faster decay would require a different method.
The paper's arithmetic formulation (Section 5) expresses the cross term through prime powers in a moving interval. This connects the two-interval criterion to the distribution of primes in short intervals, and establishing the required bound may depend on deep results beyond current knowledge, such as strong forms of the prime number theorem in very short intervals.
The paper leaves the construction of a violating pair from a hypothetical zero non-constructive (only existence is proven). Developing an explicit algorithm to compute such a pair given the zero would make the criterion more operational for disproving RH.

Original abstract

We aim to study whether the Weil form can detect a zero of the Riemann zeta function away from the critical line using two functions with increasingly narrow, widely separated supports. We prove that the Riemann hypothesis is equivalent to the Cauchy-Schwarz inequality on every such pair at all sufficiently large separations, provided the component widths decrease subexponentially. A zero to the right of the critical line forces the normalized cross term to exceed an exponential at arbitrarily large separations, with any exponent smaller than the horizontal displacement of that zero. In particular, reciprocal-width intervals suffice. Our proof combines a short-support positivity estimate, a smooth partition of fixed test functions and a pole obstruction for a one-sided Laplace transform. We also express the cross term through prime powers in a moving multiplicative interval. The resulting criterion specifies a uniform arithmetic inequality whose proof would establish the Riemann hypothesis.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-10: 1 request, 5,521 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.39 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.69 · compares with prior results 0.10 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We aim to study whether the Weil form can detect a zero of the Riemann zeta function away from the critical line using two functions with increasingly narrow, widely separated supports. We prove that the Riemann hypothesis is equivalent to the Cauchy-Schwarz inequality on every such pair at all sufficiently large separations, provided the component widths decrease subexponentially. A zero to the right of the critical line forces the normalized cross term to exceed an exponential at arbitrarily large separations, with any exponent smaller than the horizontal displacement of that zero. In particular, reciprocal-width intervals suffice. Our proof combines a short-support positivity estimate, a smooth partition of fixed test functions and a pole obstruction for a one-sided Laplace transform. We also express the cross term through prime powers in a moving multiplicative interval. The resulting criterion specifies a uniform arithmetic inequality whose proof would establish the Riemann hypothesis.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We aim to study whether the Weil form can detect a zero of the Riemann zeta function away from the critical line using two functions with increasingly narrow, widely separated supports.	contribution 0.70	0.34	0.04	unclear 0.47
s1	We prove that the Riemann hypothesis is equivalent to the Cauchy-Schwarz inequality on every such pair at all sufficiently large separations, provided the component widths decrease subexponentially.	result 0.97	0.94	0.86	asserted 1.00
s2	A zero to the right of the critical line forces the normalized cross term to exceed an exponential at arbitrarily large separations, with any exponent smaller than the horizontal displacement of that zero.	result 0.95	0.83	0.20	asserted 1.00
s3	In particular, reciprocal-width intervals suffice.	result 0.67	0.78	0.05	asserted 0.97
s4	Our proof combines a short-support positivity estimate, a smooth partition of fixed test functions and a pole obstruction for a one-sided Laplace transform.	method 1.00	0.93	0.03	asserted 1.00
s5	We also express the cross term through prime powers in a moving multiplicative interval.	method 0.61	0.91	0.03	asserted 1.00
s6	The resulting criterion specifies a uniform arithmetic inequality whose proof would establish the Riemann hypothesis.	interpretation 0.56	0.86	0.49	asserted 0.86

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** shows the reciprocal-width geometry. The outer span determines the separation, whereas each diagonal energy is evaluated on one short interval

**Figure 2.** Analytic component widths L −1 , L −2 and exp − √ L for 2 ≤ L ≤ 25, plotted on a logarithmic vertical axis to illustrate three instances of Eq (12). 4 The criterion and quantitative violations A translated even test gives a Laplace transform whose poles retain the centered zeta zeros. This supplies the obstruction to the bound in Lemma 2. Lemma 3. We suppose z0 = ρ0 − 1/2 satisfies Rz0 > α ≥ 0. There is a real even h ∈ D such that Fh(t) := B (τth, h) fails to be O

Figures reproduced from arXiv:2611.10726 by the authors.

References

Works cited (8)

[7]

P. Danesh, “Riemann Zeros as quantum energy levels,” Jul. 2026, doi: 10.33774/ COE-2026-SCGGH

View on Pith 2026
[1]

A. Weil, “SUR LES FORMULES EXPLICITES DE LA THÉORIE DES NOM- BRES,”Mathematics of the USSR-Izvestiya, vol. 6, no. 1, p. 1, Feb. 1972, doi: 10.1070/IM1972V006N01ABEH001866

View on Pith doi:10.1070/im1972v006n01abeh001866 1972
[2]

E. C. Titchmarsh and D. R. Heath-Brown, “THE THEORY OF THE RIEMANN ZETA- FUNCTION SECOND EDITION REVISED BY CLARENDON PRESS·OXFORD 1986”. 8

View on Pith 1986
[3]

H. Yoshida, “On Hermitian Forms attached to Zeta Functions,” https://doi.org/10.2969/aspm/02110281, vol. 21, pp. 281–326, Jan. 1992, doi: 10.2969/ASPM/02110281

View on Pith doi:10.2969/aspm/02110281 1992
[4]

Bombieri, Enrico. "Remarks on Weil’s quadratic functional in the theory of prime numbers, I."Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni11.3 (2000): 183-233

View on Pith 2000
[5]

A. Connes and C. Consani, “Weil positivity and trace formula the archimedean place,” Selecta Mathematica 2021 27:4, vol. 27, no. 4, pp. 77-, Jul. 2021, doi: 10.1007/ S00029-021-00689-4

View on Pith 2021
[6]

M. Suzuki, “Aspects of the screw function corresponding to the Riemann zeta-function,” J. Lond. Math. Soc., vol. 108, no. 4, pp. 1448–1487, Oct. 2023, doi: 10.1112/JLMS.12785; JOURNAL:JOURNAL:14697750;WGROUP:STRING:PUBLICATION

View on Pith doi:10.1112/jlms.12785 2023
[8]

Bombieri, E., & Lagarias, J. C. (1999). Complements to Li’s Criterion for the Riemann Hypothesis.Journal of Number Theory,77(2), 274–287. https://doi.org/10.1006/JNTH. 1999.2392. Polytechnic of Turin Payam.danesh@polito.it 9

View on Pith doi:10.1006/jnth 1999

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [7]

P. Danesh, “Riemann Zeros as quantum energy levels,” Jul. 2026, doi: 10.33774/ COE-2026-SCGGH

View on Pith 2026

[2] [1]

A. Weil, “SUR LES FORMULES EXPLICITES DE LA THÉORIE DES NOM- BRES,”Mathematics of the USSR-Izvestiya, vol. 6, no. 1, p. 1, Feb. 1972, doi: 10.1070/IM1972V006N01ABEH001866

View on Pith doi:10.1070/im1972v006n01abeh001866 1972

[3] [2]

E. C. Titchmarsh and D. R. Heath-Brown, “THE THEORY OF THE RIEMANN ZETA- FUNCTION SECOND EDITION REVISED BY CLARENDON PRESS·OXFORD 1986”. 8

View on Pith 1986

[4] [3]

H. Yoshida, “On Hermitian Forms attached to Zeta Functions,” https://doi.org/10.2969/aspm/02110281, vol. 21, pp. 281–326, Jan. 1992, doi: 10.2969/ASPM/02110281

View on Pith doi:10.2969/aspm/02110281 1992

[5] [4]

Bombieri, Enrico. "Remarks on Weil’s quadratic functional in the theory of prime numbers, I."Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni11.3 (2000): 183-233

View on Pith 2000

[6] [5]

A. Connes and C. Consani, “Weil positivity and trace formula the archimedean place,” Selecta Mathematica 2021 27:4, vol. 27, no. 4, pp. 77-, Jul. 2021, doi: 10.1007/ S00029-021-00689-4

View on Pith 2021

[7] [6]

M. Suzuki, “Aspects of the screw function corresponding to the Riemann zeta-function,” J. Lond. Math. Soc., vol. 108, no. 4, pp. 1448–1487, Oct. 2023, doi: 10.1112/JLMS.12785; JOURNAL:JOURNAL:14697750;WGROUP:STRING:PUBLICATION

View on Pith doi:10.1112/jlms.12785 2023

[8] [8]

Bombieri, E., & Lagarias, J. C. (1999). Complements to Li’s Criterion for the Riemann Hypothesis.Journal of Number Theory,77(2), 274–287. https://doi.org/10.1006/JNTH. 1999.2392. Polytechnic of Turin Payam.danesh@polito.it 9

View on Pith doi:10.1006/jnth 1999