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Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that the Alternative Hypothesis, the main rival to the standard pair correlation conjecture, also forces asymptotically 100% of the zeros of the Riemann zeta-function to be simple and on the critical line, with no…

desk verdict A careful, honest conditional paper: the 100% simplicity result is real but rests on a new model axiom (AH2) that does the decisive work; still deserves a serious referee. read the letter →

arxiv 2507.06823 v1 pith:7SKPUBSC submitted 2025-07-09 math.NT

Daniel A. Goldston , Junghun Lee , Jordan Schettler , Ade Irma Suriajaya This is my paper · ORCID
classification math.NT MSC 11M0611M26
keywords Riemannzeta-functionzerospaircorrelationAlternativeHypothesissimplecriticallineEssentialSimplicitysecondmoment
topics The Riemann Hypothesis
open problems The Riemann Hypothesis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to show that a conclusion previously drawn from the standard pair correlation conjecture — that asymptotically 100% of the nontrivial zeros of the Riemann zeta-function are simple and lie on the critical line — is not an accident of that one conjecture. Under the Alternative Hypothesis, the main competing picture of how zeros are spaced, it formulates a precise 'weak density' version and proves the same 100% conclusion, without assuming the Riemann Hypothesis. The load-bearing step is a weighted average identity (Theorem 2) that links the pair densities $P_{k/2}(T)$ to the second moment of zeros in short intervals; adding a new half-integer density sum rule forces $p_0 = 1$, which is equivalent to Essential Simplicity. If correct, the simplicity question is settled under either competing conjecture and the Riemann Hypothesis is not required.

What carries the argument

The central objects are the pair densities $P_{k/2}(T)$, defined as the number of pairs of zeros $0<\gamma,\gamma'\le T$ whose normalized distance $(\gamma-\gamma')L$ lies in $(k/2-1/4,\, k/2+1/4]$, divided by $TL$. The argument runs on a weighted identity (Theorem 2) obtained by equating an unconditional second-moment asymptotic for zeros in short intervals with a combinatorial expansion of the same weighted count under AH-Pairs. The new auxiliary input AH-Weak Density (conditions AH1 and AH2) then turns this identity into asymptotic formulas for $P_0(T)$, yielding $p_0=1$. The half-integer sum rule AH2 is the specific new mechanism that fixes the average of the half-integer densities to $M/2 - 1/4$.

What would settle it

Compute, from the first several million zeros of $\zeta(s)$, the densities $P_{j-1/2}(T)$ for $1\le j\le M$ with $M$ large, and check whether their average minus $M/2$ approaches $-1/4$ within the stated error $O(1/M)$; a systematic deviation would refute AH2 and break Theorem 4. Alternatively, a direct numerical estimate of $p_0$ differing from 1 at a scale not explained by the error terms would falsify the conclusion.

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Extended reading notes

Core claim

The paper's central claim is that, under the Alternative Hypothesis for pairs of zeros (AH-Pairs) together with the newly stated Alternative Hypothesis for Weak Density (AH-Weak Density), the limiting density $p_0$ of pairs of zeros within a quarter of the average spacing equals 1. By Theorem 1, $p_0 = 1$ is equivalent to Essential Simplicity, and Essential Simplicity is already known to imply that asymptotically 100% of zeros are simple and on the critical line. The proof does not invoke the Riemann Hypothesis at any point; RH is used only to motivate the formulation. The auxiliary hypothesis splits into (AH1), a fixed asymptotic value for the sum of adjacent half-integer and integer pair densities, and (AH2), a half-integer density sum rule giving average $M/2 - 1/4$. The latter is the only new input needed to pass from the average relations of Theorem 3 to $p_0=1$.

Load-bearing premise

The paper's conclusion depends on the auxiliary conjecture AH2, which asserts that the average of the first $M$ half-integer pair densities tends to $M/2 - 1/4$; this is not derived from AH-Pairs and is the new input that forces $p_0 = 1$.

Editorial extensions

If this is right

  • Under AH-Pairs and AH-Weak Density, asymptotically 100% of the zeros of $\zeta(s)$ are simple and lie on the critical line, with no Riemann Hypothesis assumption (Theorem 4).
  • Assuming AH-Pairs, $p_0=1$ and Essential Simplicity are equivalent (Theorem 1), so the 100% conclusion is interchangeable with the pair-repulsion and simplicity statement (ES).
  • If only AH1 is assumed, the weaker conclusion holds: at least 50% of zeros are simple and at least 50% are on the critical line, with $\limsup_{T\to\infty} P_0(T) \le 3/2$ (Corollary 2).
  • Under RH, $p_0=1$ together with the known density relations determines all limiting densities $p_{k/2}$: $1/2$ for nonzero even $k/2$ and $1/2 - 2/(\pi^2 k^2)$ for odd $k/2$ (Equation (1.17)).

Reading between the lines

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

  • We would add that AH2 is the real new hypothesis: it is not derived from AH-Pairs, and the paper's 100% conclusion collapses to the weaker 50% bounds if AH2 fails, so the credibility of the theorem rests on this specific average rule.
  • Going beyond the paper, the same weighted second-moment technique could be applied to other $L$-functions or to the derivative of $\zeta$, where the Alternative Hypothesis has also been studied; one would expect an analogous half-integer density sum rule to control the proportion of simple zeros there.
  • We infer that if the 100% conclusion holds under both the standard pair correlation conjecture and the Alternative Hypothesis, the simplicity and on-line properties of zeta zeros may be insensitive to the fine details of vertical pair correlation, pointing toward a model-independent theorem that would hold under any pair-correlation conjecture of the same shape.
  • Numerically, one can test AH2 at finite $T$ with the first few million zeros: the prediction that the average of the first $M$ half-integer densities is $M/2 - 1/4$ is precise enough to be distinguished from the standard pair-correlation prediction, which behaves differently.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies the implications of the Alternative Hypothesis (AH) for the simplicity and critical-line placement of zeros of the Riemann zeta-function, without assuming RH. It introduces two hypotheses: AH-Pairs, a version of AH asserting that pair spacings are close to half-integers, and a new AH-Weak Density consisting of two axioms, (AH1) and (AH2), which specify averages of the densities P_{j-1/2} and P_j. Theorem 1 shows that, under AH-Pairs, p0=1 is equivalent to Essential Simplicity. Theorem 2 is a conditional identity for the Gallagher-Mueller second-moment sum. Combining Theorem 2 with (AH1) gives Theorem 3, and then adding (AH2) yields p0=1 in Theorem 4, hence 100% simple and critical zeros. Without (AH2), Corollary 2 gives only limsup P0 ≤ 3/2 and at least 50% simple and critical zeros. The proofs are detailed and the use of known second-moment estimates is careful.

Significance. If AH2 were a consequence of AH-Pairs or independently well-motivated, the result would be a significant extension of the Gallagher-Mueller method beyond Montgomery's pair correlation conjecture, showing that a different pair-correlation model also forces essential simplicity. The paper is transparent about its conditional structure and correctly uses unconditional estimates of Gallagher-Mueller, Fujii, and Tsang. However, the central 100% conclusion is not a consequence of AH-Pairs alone: AH2 fixes the average of the half-integer densities and, via Theorem 3, directly forces p0=1. The RH-conditional family (1.15) shows that models with p0 in the entire allowed interval [1, 3/2 - 2/pi^2] satisfy AH-Pairs and (AH1), but AH2 holds only at p0=1. Thus the paper's main theorem is essentially an implication from an axiom chosen to encode the conclusion, rather than a demonstration that the Alternative Hypothesis itself implies essential simplicity. The 50% bounds of Corollary 2 are the strongest conclusions that do not rely on AH2.

major comments (2)
  1. [Section 1, definition of AH-Weak Density] The load-bearing step toward p0=1 is the new axiom AH2, not AH-Pairs. Indeed, substituting AH2 into (1.22) gives P0(T) = 1 + O(sqrt(log M)/M) + O(M(R(T)+RP(T)+1/L^2)), so Theorem 4's conclusion follows immediately. Conversely, under AH-Pairs and (AH1), equation (1.22) shows that p0=1 is equivalent to the averaged half-integer density tending to 1/2, which is exactly what AH2 asserts up to its stated errors. The paper even displays the RH-conditional family (1.15): for every p0 in [1, 3/2 - 2/pi^2] the limiting densities satisfy AH-Pairs, (AH1), nonnegativity, and the bound (1.13), but AH2 holds only at p0=1. Therefore the 100% result is effectively an input of the model, and without AH2 the upper-endpoint model p0=3/2 - 2/pi^2 is not eliminated. The abstract and introduction should state clearly that the 100% conclusion is conditional on the additional AH2 conjecture, and the paper should discuss what independent evidence, if any, supports AH2.
  2. (AH1) is also an additional strengthening of AH-Pairs, not a consequence of it. AH-Pairs gives errors of size O((|k|+1)R(T)) for the location of individual pair spacings, while (AH1) asserts a uniform, j-independent error O(RP(T)) for the sums P_{j-1/2}(T)+P_j(T). The footnote in the paper explains this choice heuristically, but the statement 'AH1 is obtained immediately from (1.15)' may mislead: (1.15) is a limiting formula under RH, not an error estimate under AH-Pairs. The paper should explicitly label (AH1) and (AH2) as new model assumptions distinct from AH-Pairs, and should justify or at least clearly flag the strengthening involved in making the error independent of j.
minor comments (5)
  1. [Equation (1.18)] The error term appears as 'O(p log M)' in the displayed equation; this should be O(sqrt(log M)) as written in (1.22).
  2. [Equation (1.13)] The displayed chain contains a duplicated summation symbol; the intended statement is simply that |P(T,M)| ≪ M TL, hence the sum of the densities is O(M).
  3. [Section 1, AH2] AH2 is stated as uniform in M on compact intervals, but the derivation of p0=1 passes to the limit M→∞ for fixed T and then lets T→∞. The sentence should be clarified so that the order of limits is explicit and no growth of M with T is required.
  4. [Section 2, equation (2.2)] The notation N(T, C0 R(T)) is used without re-stating that it counts pairs with |(gamma-gamma')L| ≤ C0 R(T); the symmetric version of N(T, lambda) should be defined or explicitly recalled here.
  5. [General formatting] There are several spacing/OCR artifacts in the title and running heads ('P AIR CORRELA TION', 'W eak', 'F aculty') that should be corrected in the final version.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 4's p0=1 conclusion is selected by the newly introduced AH2 sum rule rather than derived from AH-Pairs; the 100% claim is accordingly an input of the model.

  1. self definitional [Section 1, AH-Weak Density (AH2), Theorem 3 equations (1.22)-(1.24), Theorem 4]
    "If in Theorem 3 we also assume (AH2), then by using either (AH2) in (1.22) or (1.21) in (1.23), we immediately obtain (1.24) P0(T ) = 1 + O( sqrt log M / M ) + O( M( R(T ) + RP (T ) + 1/L^2 ) ). Thus p0 = lim_{T→∞} P0(T ) = 1 + O( sqrt log M / M ),"

    The new axiom (AH2) fixes the average half-integer density to M/2 - 1/4. Under the RH-conditional model (1.15)-(1.17), that exact value is the sum obtained only when p0 = 1. Inserting (AH2) into (1.22) forces P0(T) = 1 in (1.24), which is the central conclusion of Theorem 4. Thus the target value p0 = 1 is already encoded in the newly introduced sum rule (AH2); it is not derived from AH-Pairs. Without (AH2) the paper retains only Corollary 2's weaker limsup P0 ≤ 3/2 and 50% bounds.

full rationale

The paper is mathematically transparent: Theorem 1 is proved from AH-Pairs alone, and Theorem 2 is a real Gallagher-Mueller averaging result independent of the new axioms. The collapse to p0 = 1, however, occurs exactly at the point where (AH2) is substituted into Theorem 3: (AH2) asserts the half-integer average that the p0 = 1 branch of the RH-conditional formula (1.17) would produce, and (1.24) then returns P0(T) = 1. So the headline 100% conclusion of Theorem 4 is a consequence of a new axiom chosen to produce it, not a consequence of AH-Pairs alone. The paper does not hide this—AH-Weak Density is labeled a model—but the central claim reduces, by construction, to that model assumption. The same-author citations to [GLSS25] for the ESH-to-100% step are load-bearing but contain independent mathematical content, so they are not separately counted as circular. Overall, partial circularity: the derivation supplies real intermediate theorems, but the decisive p0 = 1 result is an input of the AH2 model rather than an output of the original AH-Pairs framework.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The paper's central claim rests on standard analytic number theory tools plus two layers of conjecture: the previously studied AH-Pairs and the new AH-Weak Density (AH1, AH2). The latter is the main input the reader does not get from prior literature, and the conclusion p0 = 1 is essentially a consequence of AH2.

free parameters (2)
  • R(T) = not fitted; assumed positive decreasing to 0
    Introduced in AH-Pairs (AH0), Section 1, as the error rate for how closely pair spacings approach half-integers. It is part of the assumed conjecture, not fitted to data.
  • RP(T) = not fitted; assumed positive decreasing to 0
    Introduced in AH-Weak Density (AH1, AH2), Section 1, as the error rate for the density sum rules. It is part of the new conjecture.
assumptions (6)
  • domain assumption AH-Pairs: for all pairs in P(T,M), (gamma-gamma')L = k/2 + O((|k|+1)R(T)) for some integer k, with R(T) -> 0.
    Assumed throughout; defined in Section 1, equation (AH0). This is the core alternative pair correlation conjecture.
  • ad hoc to paper AH1: P_{j-1/2}(T) + P_j(T) = 1 - 2/(pi^2(2j-1)^2) + O(RP(T)) for each j.
    Formulated in Section 1; mirrors the RH-conditional formula (1.15) but is used here as an unproved assumption in the new weak density conjecture.
  • ad hoc to paper AH2: sum_{j=1}^M P_{j-1/2}(T) = M/2 - 1/4 + O(1/M) + O(M RP(T)).
    Formulated in Section 1; this is the load-bearing new condition that forces p0 = 1 via Theorem 3.
  • domain assumption Gallagher-Mueller second moment identities (Proposition 1 and Proposition 2 from [GLSS25, Section 2]).
    Used as black-box results to evaluate D(T,M) in Section 4; proven in the companion paper [GLSS25].
  • standard math Unconditional pair-counting bound (GM87, Lemma 9): sum over pairs with |(gamma-gamma')L| <= h is O((1+h) T L).
    Used in (1.12), (2.1), and (4.1) to bound density sums.
  • standard math Standard zero counting and Riemann-von Mangoldt formula: N(T) ~ T L and the S(T) estimates.
    Used throughout, e.g., (1.1) and Proposition 2.
invented entities (1)
  • AH-Weak Density (AH1 and AH2)
    purpose: New auxiliary conjecture specifying density sum rules for pairs of zeros near half-integer spacings; it enables the proof of p0 = 1 and essential simplicity from AH.
    The conjecture is introduced in this paper in Section 1. It has a falsifiable nature (it could be checked numerically), but no independent evidence is presented and it is not derived from AH-Pairs.

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Cite this review

Pith. "Pith review of Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis." pith.science (2026). https://pith.science/paper/7SKPUBSC

@misc{pith2026250706823,
  author       = {Pith},
  title        = {Pith review of: Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7SKPUBSC}},
  note         = {Machine review of arXiv:2507.06823}
}
read the original abstract

In an earlier paper, we proved that Montgomery's Pair Correlation Conjecture (PCC) for zeros of the Riemann zeta-function can be used to prove without the assumption of the Riemann Hypothesis (RH) that asymptotically 100% of the zeros are both simple and on the critical line. This is based on a method of Gallagher and Mueller from 1978. We formulate an appropriate form of the Alternative Hypothesis (AH), which determines a different PCC, and, using the same method as above, prove that asymptotically, 100% of the zeros are both simple and on the critical line. As in our previous paper, we do not assume RH.

Discussion (0). Continue with ORCID to comment.

Reference graph

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