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Beyond the Riemann Hypothesis bounds: A pair-correlation approach to the least prime in arithmetic progression and the smallest quadratic non-residue

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Pith's one-line read Assuming a vertical zero-spacing hypothesis for Dirichlet L-functions, the paper pushes the least quadratic non-residue down to (log q)^{1+ε} and the least prime in an arithmetic progression down to φ(k) times a subexponential factor.

desk verdict Original conditional results, but the proof of the main theorems has two concrete gaps that need fixing before the claims can be trusted. read the letter →

arxiv 2607.14515 v1 pith:NYTE6BXE submitted 2026-07-16 math.NT

Neelam Kandhil , Alessandro Languasco , Pieter Moree This is my paper · ORCID
classification math.NT MSC 11M2611N1311L4011M50
keywords leastquadraticnon-residueprimesinarithmeticprogressionspaircorrelationDirichletL-functionsgeneralizedRiemannhypothesiszerospacingLinnik'sconstantMontgomerypair-correlationconjecture
topics 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

The paper asks whether the vertical spacing—not just the horizontal location—of the nontrivial zeros of Dirichlet L-functions can improve the classical conditional bounds for two famous problems. Its central claim is that if a natural pair-correlation bound F^+ ≪ T log(kx) holds down to heights T as small as x^ε (Hypothesis 1), then the least quadratic non-residue modulo a prime q satisfies n(q) ≪ (log q)^{1+ε}, beating the long-standing GRH bound (log q)^2. Symmetrically, if a related pair-correlation estimate holds (Hypothesis 2), then the least prime in an arithmetic progression satisfies p(k) ≪ φ(k) exp(B (log k)^A) for some A ∈ (1/2,1), improving on the previous k^{1+ε} under the same framework. Both results are conditional on the Generalized Riemann Hypothesis and on pair-correlation hypotheses that the paper formulates but does not prove; the unconditional part of the paper is a Montgomery-type bound for F^+ in the range x ≤ T ≤ e^x. A sympathetic reader will see the paper as establishing a bridge: vertical zero repulsion, if confirmed, converts a spectral regularity into sharp arithmetical bounds.

What carries the argument

The load-bearing object is the pair-correlation sum F^+_{χ_□}(x,T) (and its full-sum analogue F_k(x,T) for characters mod k), which measures how zeros repel: it is essentially the Fourier transform of a quadratic character's zero-count, weighted by W(u)=4/(4+u²). Montgomery's 1973 argument gives a bound in the range T ≥ x; the paper's innovation is to treat the short-T range as a hypothesis and push it through an explicit-formula dyadic argument, converting the spectral input into bounds for n(q) and p(k). The Sobolev–Gallagher inequality and a Goldston–Montgomery Fourier-series lemma are the technical tools that let the authors pass from the pair-correlation bound to estimates on weighted r

What would settle it

Compute F^+_{χ_□}(x,T) for a fixed small prime modulus (say q ≈ 10^6) at heights in the gap between x^ε and x; if for some T in that range F^+/(T log(qx)) exceeds a constant, then Hypothesis 1 fails and Theorem 3 has no content. Also, deriving a contradiction from the short-interval prime number theorem used at Eq. (32) would invalidate Theorem 1's uniformity range.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the pair-correlation function F^+_{χ_□}(x,T), which sums x^{i(γ_1−γ_2)} over pairs of zeros of the quadratic character L-function weighted by W(γ_1−γ_2)=4/(4+(γ_1−γ_2)^2), satisfies F^+ ≪ T log(kx) for x ≤ T ≤ e^x (Theorem 1) and, under GRH, has the Montgomery-type asymptotic F^+ ∼ (T/2π) log x (Theorem 2). The paper then postulates (Hypothesis 1) that the same bound persists for T as small as x^ε, and uses that extension—via a dyadic decomposition of the explicit formula for θ(x,χ_□)—to prove n(q) ≪ (log q)^{1+ε} (Theorem 3). For primes in arithmetic progressions, the analogous Hypothesis 2 yields, under GRH, a Chebyshev-type error ψ(x

Load-bearing premise

The whole improvement for the least quadratic non-residue rests on Hypothesis 1—that the pair-correlation bound F^+ ≪ T log(kx) continues to hold when the height T drops from x down to x^ε—while the proof only establishes this bound for T ≥ x.

Editorial extensions

If this is right

  • Under RH plus Hypothesis 1, n(q) ≤ (log q)^{1+ε} for large primes q, nearly attaining Vinogradov's conjecture n(q) ≪ q^ε.
  • If the T-range in Hypothesis 1 can be extended to (log x)^2, the bound refines to n(q) ≪ (log q)(log log q)^2.
  • Under GRH plus Hypothesis 2, p(k) ≪ φ(k) exp(B (log k)^A) with A < 1, a subexponential bound that improves on p(k) < k^{1+ε}; it remains far from Heath-Brown's conjectured k(log k)^2.
  • The same hypothesis yields uniform Chebyshev-error equidistribution: ψ(x;k,a) − x/φ(k) ≪ √x/φ(k) exp(c_1 (log x)^{c_2}) for k up to x exp(−2c_1(log x)^{c_2}).
  • Theorem 2 gives the Montgomery-type asymptotic F^+ ∼ (T/2π) log x under GRH in the range x ≤ T ≤ e^x, evidence that quadratic-character zeros behave like the zeta zeros.

Reading between the lines

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

  • The real bottleneck is Hypothesis 1: Theorem 1 only proves F^+ ≪ T log(kx) for T ≥ x, and the entire improvement from (log q)^2 to (log q)^{1+ε} rests on the unproved extension into T ≥ x^ε. A numerical check of F^+ in that short-T range for a small prime modulus would be a cheap way to test whether the hypothesis is plausible.
  • A subtle gap sits inside the proof of Theorem 1: the estimate at Eq. (32), which uses a short-interval prime number theorem, is not derived from the stated RH assumption. If that step fails, the uniformity range of the base pair-correlation bound—and hence the input to Hypothesis 1—would need revisiting.
  • If these hypotheses are ever proved, the method would likely yield similar sharpening for other GRH-limited problems, such as the Pólya–Vinogradov bound on character sums, by the same 'vertical repulsion' mechanism.
  • The Friedlander–Granville limitations quoted in the paper show that the uniformity range in Theorem 4 is essentially maximal; the subexponential term cannot be replaced by a power of log x, so the paper's bounds are near the boundary of what is possible.
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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

3 major / 4 minor

Summary. The paper proposes two conditional improvements under pair-correlation hypotheses for Dirichlet L-functions. Under RH for a quadratic character plus a postulated bound (Hypothesis 1) for F^+_{χ_q}(x,T) in the range x^ε ≤ T < x, Theorem 3 claims n(q) ≪ (log q)^{1+ε}. Under GRH plus a second pair-correlation hypothesis (Hypothesis 2), Theorems 4 and 5 claim a uniform error bound ψ(x;k,a) − x/φ(k) ≪ √x exp(c_1(log x)^{c_2})/φ(k) for k ≤ x exp(−2c_1(log x)^{c_2}), and consequently p(k) ≪ φ(k) exp(2c_1(log k)^{c_2}) with c_2 ∈ (1/2,1). The paper also proves Theorem 1, an upper bound for F^+ in the range x ≤ T ≤ e^x, and Theorem 2, an asymptotic in a restricted range.

Significance. If the conditional results were correct, they would constitute notable improvements over the classical GRH bounds (Ankeny, Bach–Sorenson, Lamzouri–Li–Soundararajan), and the proposed connection between zero pair-correlation and these two classical sieving problems is conceptually attractive. The main conditional theorem for quadratic non-residues (Theorem 3) appears internally coherent and is a genuine conditional implication, although it rests on the very strong, unproved Hypothesis 1. However, the second half of the paper, Theorems 4 and 5, is not merely missing a detail: Theorem 4 contradicts the Friedlander–Granville limitation quoted by the authors themselves, and the proof of Theorem 5 inherits the failure. The advertised bound p(k) ≪ φ(k) exp((log k)^A) is therefore unsupported and likely false as stated. The paper does not provide machine-checked proofs or reproducible code; its strengths are the clear structure and the standard explicit-formula framework.

major comments (3)
  1. [§5, Theorem 4, Eq. (54)] The tail term ψ(x/2^{J+1};k,a) − x/(2^{J+1}φ(k)) is bounded by (log x)^2 + x/(2^{J+1}φ(k)), and this is then absorbed into the claimed √x exp(c_1(log x)^{c_2})/φ(k). This absorption fails at the upper end of the stated k-range. Let B = c_1(log x)^{c_2} and k = x e^{−2B}. Then J=0 and the tail main-term contribution is at least x/(2φ(k)) ≥ e^{2B}/2, while the claimed error is e^{3B}/√x = o(1). Thus the displayed inequality in (54) is false for this k. This is not a minor technicality: applying the Friedlander–Granville limitation (6) with y=x/2 and the same k gives, for some a, an error ≫ exp(2(1−c_2)B), which also exceeds e^{3B}/√x. Hence Theorem 4 as stated is false.
  2. [§5, Theorem 5] The proof of Theorem 5 relies entirely on Theorem 4, so it inherits the failure described above. Independently, the positivity step in the proof is written as if x/φ(k) − D√x e^B/φ(k) > 0 implies x ≫ φ(k)e^{2B}; the direct algebra gives x ≫ e^{2B}. The factor φ(k) has to be imported from the theorem's hypothesis range k ≤ x e^{−2B}. With the correct constraint, the best conclusion one could hope for from the given method is p(k) ≪ k exp(B(log k)^A), not the claimed φ(k) version. Since Theorem 4 is false, Theorem 5 is unsupported.
  3. [§3, Lemma 6, Eq. (32)] The proof of Lemma 6 bounds E_1 by Tδx using the estimate ψ(n+10^4δx)−ψ(n) ≈ 10^4δx, described as following from the prime number theorem. With δ = √(S(x)/(Tx)) and T ≥ x, the interval length is δx ≪ √(log x). No form of the prime number theorem available under the paper's assumptions gives an asymptotic for ψ in intervals of length o(√n) uniformly for all n ≤ x; the usual error terms, whether unconditional or under RH, are far larger than the interval length. This step is unjustified, so the proofs of Lemma 6, Theorem 1, and Theorem 2 are incomplete. Theorem 3 itself does not use Theorem 1, but this gap affects a stated theorem of the paper.
minor comments (4)
  1. [§1.2, Lemma 11] The statement of Lemma 11 has a typo: the maximum should be over t, i.e. max_{U≤t≤T} F^+_k(x,t), not F^+_k(x,T).
  2. [§4, Lemma 9] The line 'Let k > xε' should read 'Let k > x^ε'.
  3. [§4, Remark 4] Remark 4 refers to Lemma 13, which is introduced later in Section 5. This is a minor organizational issue.
  4. [Abstract / Introduction] The abstract and introduction state that the paper 'surpasses' classical GRH bounds. Since the results are conditional on additional unproved Hypotheses 1 and 2, the wording should be qualified (e.g., 'conditional on a pair-correlation hypothesis').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: main theorems are explicit conditional deductions from pair-correlation hypotheses stated in zero statistics, not in terms of n(q) or p(k).

full rationale

The paper's central results (Theorems 3, 4, and 5) are conditional on Hypotheses 1 and 2, which are formulated directly in terms of the zero-pair-correlation functions F^+_χ□(x,T) and F_k(x,T). These hypotheses do not mention n(q) or p(k), and the path from them through explicit formulas, Lemmas 7–9 and 11–13, and dyadic decomposition is a genuine derivation rather than a relabeling or a fitted-parameter prediction. The self-citations to the authors' [13] supply auxiliary lemmas and context, but they do not constitute the load-bearing premises that generate the new estimates; hence there is no circular dependency. The manuscript even flags its own limitation in Remark 2, noting that the technique does not approach Heath-Brown's conjecture because the required k-uniformity would contradict Friedlander–Granville. The serious concerns in this paper are correctness gaps, not circularity: Eq. (32) uses a short-interval PNT step ("E1 ≪ ... ≪ Tδx ... where we have used the prime number theorem in the final step") that is not justified by the stated assumptions, and Eq. (54) appears to absorb a tail x/(2^{J+1}φ(k)) into O(√x e^B/φ(k)) that fails near k = x exp(−2B). These would affect validity of Theorems 1 and 4, but they are not instances of an input being made equivalent to an output by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's conditional results rest on (i) standard GRH-type assumptions, (ii) two newly stated pair-correlation hypotheses (Hypotheses 1 and 2) which are natural analogues of Montgomery's conjecture but unproved and tailored to the target bounds, and (iii) a standard analytic number theory toolbox. No fitted parameters or invented entities are present; the constants ε, c1, c2 are universal/existential parameters, not fitted.

assumptions (5)
  • domain assumption GRH for Dirichlet L-functions (all characters mod k)
    Assumed in Theorems 2, 4, 5 and Hypothesis 2; unproved.
  • domain assumption RH for quadratic Dirichlet L-functions L(s,χ_□)
    Assumed in Theorems 1, 3 and Hypothesis 1; unproved.
  • ad hoc to paper Hypothesis 1: F^+_{χ_□}(x,T) ≪ T log(kx) uniformly for x^ε ≤ T < x and 3≤k≤exp(x(log x)^{-1})
    New pair-correlation conjecture introduced to derive n(q) ≪ (log q)^{1+ε}; analogue of Montgomery's conjecture but unproved and unverified in this range.
  • ad hoc to paper Hypothesis 2: F_k(x,T), F^+_k(x,T) ≪ φ(k) T exp(c1 (log x)^{c2}) for exp(c1(log x)^{c2}) ≤ T < x and exp(...) < k ≤ x exp(-2c1(log x)^{c2})
    New assumption used for Theorems 4-5; its k-uniformity is exactly at the Friedlander–Granville limitation.
  • standard math Classical analytic number theory toolkit (explicit formulas Lemmas 1-2, Brun–Titchmarsh, Sobolev–Gallagher, Goldston–Montgomery Lemma 5, Cauchy–Schwarz, PNT, orthogonality of characters)
    Used throughout proofs; standard background not proved in the paper.

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Pith. "Pith review of Beyond the Riemann Hypothesis bounds: A pair-correlation approach to the least prime in arithmetic progression and the smallest quadratic non-residue." pith.science (2026). https://pith.science/paper/NYTE6BXE

@misc{pith2026260714515,
  author       = {Pith},
  title        = {Pith review of: Beyond the Riemann Hypothesis bounds: A pair-correlation approach to the least prime in arithmetic progression and the smallest quadratic non-residue},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NYTE6BXE}},
  note         = {Machine review of arXiv:2607.14515}
}
read the original abstract

The Generalized Riemann Hypothesis (GRH) has long defined the expected bounds for the smallest prime in an arithmetic progression and the least quadratic non-residue. However, this hypothesis primarily addresses the horizontal location of non-trivial zeros. In this paper, we show that incorporating the vertical spacing--or pair-correlation--of these zeros allows us to surpass these classical bounds. By combining these two zero-distribution perspectives, we establish sharper estimates for both problems under GRH and specific pair-correlation hypotheses, thereby providing a new link between pair-correlation phenomena for Dirichlet L-functions and these two classical problems.

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Reference graph

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