Pith. sign in

REVIEW 3 major objections 5 minor 15 references

Levinson's theorem and its generalization for Dirichlet L-functions

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read At least one-third of zeta's non-trivial zeros, and more than 41.72% of Dirichlet L-function zeros, lie on the critical line; more than 40.74% of non-trivial zeros are simple and on the line.

desk verdict An honest, clearly written student report that reproduces known theorems by Young and Wu, with no new mathematics and a load-bearing error-term estimate deferred to Wu's paper. read the letter →

arxiv 2511.06109 v2 pith:ZBFBIAKP submitted 2025-11-08 math.NT

Swapnil Ray This is my paper · ORCID
classification math.NT MSC 11M2611M06
keywords Levinson'stheoremmethodRiemannzetafunctionDirichletL-functionsmollifiedsecondmomentsimplezeroscriticallinetwistedmeansquare
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

This report sets out to show that Levinson's method — converting a mean value of an L-function against a short Dirichlet polynomial, called a mollifier, into a lower-bound count of zeros on the critical line — proves two positive-proportion theorems: at least one-third of the non-trivial zeros (those in the critical strip) of the Riemann zeta function lie on Re(s) = 1/2, and for any Dirichlet L-function with conductor growing slowly relative to height, more than 41.72% of non-trivial zeros lie on the critical line, with more than 40.74% of them both simple and on the line. The gain over the one-third bound comes from lengthening the mollifier to T^(4/7−ε), the longest range for which the needed error-term estimate is claimed, then optimizing auxiliary polynomials; longer mollifiers detect more zeros. These are unconditional statements toward the Riemann and generalized Riemann hypotheses — no unproved hypothesis is assumed. The report follows a 2010 short proof of Levinson's theorem and a 2018 generalization for Dirichlet L-functions, with the heaviest estimates cited from the literature rather than re-derived.

What carries the argument

The load-bearing object is the mollified second moment I_R(Q, χ) = ∫_T^(2T) |V(1/2 + it + R/L_χ, χ)|² |B(1/2 + it, χ)|² dt, where B(s, χ) = Σ_(n≤y) χ(n)a(n)n^(−s) is the mollifier (a short Dirichlet polynomial of length y = T^θ) and V is the L-function acted on by Q(−(1/L_χ)d/ds). Levinson's inequality, κ(χ) ≥ 1 − (1/R) log(T^(−1)I_R(Q, χ)) + o(1), converts an asymptotic for this moment into a zero-proportion bound. The essential input is Theorem 3.2, which evaluates I_R(Q, χ) up to an error T^(1−ε0), ε0 > 0, for θ < 4/7 when the coefficients have the form a(n) = μ(n)(F0 + F1·(F2*F3))(n) with separable factors F_i; this longer-mollifier regime is what pushes the proportion past two-fifths, a

What would settle it

Two checks settle it. (1) Reproduce the deferred estimate of §III.2 — Z ≪_ε y^(7/8)T^(−1/2+11η/2+ε) + y^(7/4)T^(−1+11η/2+ε) — and confirm it holds uniformly for θ < 4/7 with a(n) = μ(n)(F0 + F1·(F2*F3))(n); any θ ≥ 4/7 failure would break Theorem 3.1. (2) Recompute the numerical step: insert the listed polynomials P1, P2, P, Q into the formula for c(P, Q, R, θ) and verify κ ≥ 1 − (1/R) log c gives 0.4172 (and 0.4074 for the linear Q with R = 1.116); a smaller result would refute the constants.

Watch

Extended reading notes

Core claim

Theorem 3.1 asserts that for any Dirichlet character χ, with log q = o(log T), the proportion κ(χ) of non-trivial zeros of L(s, χ) on the critical line exceeds 0.4172 for large T, and the proportion κ*(χ) of zeros both on the line and simple exceeds 0.4074. This generalizes the 1989 two-fifths result for zeta, and the report also proves Levinson's theorem: at least one-third of zeta's non-trivial zeros lie on the line. Both rest on one mechanism: Theorem 3.2, an asymptotic for the mollified second moment with error T^(1−ε0) (ε0 > 0) valid for mollifier length θ = 4/7 − ε under a special coefficient shape, converted by Levinson's inequality into a zero count. The longer mollifier — beyond Lev

Load-bearing premise

The load-bearing premise is that the error term in the mollified second-moment asymptotic (Theorem 3.2) is genuinely small — T^(1−ε0) with ε0 > 0 — all the way to mollifier length θ = 4/7 − ε for the special coefficient shape a(n) = μ(n)(F0 + F1·(F2*F3))(n); the report defers this estimate to the cited literature, and if that bound's range or exponent is wrong, the constants 0.4172 and 0.4074 do not follow from anything shown here.

Editorial extensions

If this is right

  • Unconditional positive proportions: at least one-third of zeta's non-trivial zeros, more than 41.72% of Dirichlet L-function zeros, and more than 40.74% of zeros simple and on the line — all without assuming the Riemann hypothesis or any unproved input.
  • The same template transfers: each historical extension of the admissible mollifier length (the report surveys θ < 17/33 and θ < 6/11, each raising the zeta proportion) goes through the same Theorem-3.2-plus-Levinson-inequality route, so any future error-term improvement directly upgrades the constants.
  • The numerical step is explicit and reproducible: with θ = 4/7 − ε, R = 1.3 and the listed Q, P1, P2, P, the formula for c(P, Q, R, θ) gives κ > 0.4172; with the linear Q and R = 1.116 it gives κ* > 0.4074.
  • Simplicity comes from the same inequality: when Q is a linear polynomial, Levinson's inequality bounds the simple-and-critical proportion κ*(χ), so the simplicity statement is a by-product of the same moment evaluation, not a separate argument.

Reading between the lines

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

  • Editorial inference: the proof as written is a condensation — §III.2 states that the decisive error estimate 'is quite lengthy and is given in more details in [13]', and the sieve evaluation of Σ(α, β) is quoted without proof — so a reader seeking the constants 0.4172/0.4074 fully verified from this report alone must supply those deferred arguments.
  • Editorial inference: the report's own conclusion cites later results pushing the Dirichlet bounds well above two-fifths (roughly 0.59 for critical zeros, over 0.60 for simple critical zeros), so the report is best understood as an expository reconstruction of the longer-mollifier mechanism and its numerical optimization rather than a new record.
  • Editorial inference (testable extension): the constants depend on a numerical optimization of P1, P2, P, Q at fixed θ = 4/7 − ε; re-running that optimization with higher-degree polynomials or a swept range of R would reveal how much headroom the same error-term range still contains, without any new analytic input.
  • Editorial inference: because the report's preliminaries set up the method for general L-functions (approximate functional equation, completed L-function), the same two-step recipe — a twisted-moment asymptotic plus Levinson's inequality — would give an analogous zero-proportion statement for any other L-function family whose moment asymptotic can be established.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This manuscript is an expository report that first sketches Matthew Young's proof of Levinson's theorem for the Riemann zeta function (at least one-third of non-trivial zeros lie on the critical line) and then claims a generalization to Dirichlet L-functions: Theorem 3.1 states that for any Dirichlet character χ and sufficiently large T with log q = o(log T), κ(χ) > 0.4172 and κ*(χ) > 0.4074, where κ and κ* are the proportions of critical zeros and simple critical zeros, respectively. The proof of the Dirichlet part follows Xiaosheng Wu's 2018 paper, using a mollified second moment and a long-mollifier device. The manuscript is not a research announcement of a new theorem; it is a survey based on the cited works [7], [10], and [13], with explicit choices of polynomials and numerical parameters. The central technical claim is Theorem 3.2, an asymptotic for the mollified second moment I_R(Q,χ), and Section III.3 uses it together with quoted Levinson-type inequalities and a sieve estimate to obtain the numerical bounds.

Significance. If the exposition were fully self-contained and accurate, the paper would be a useful pedagogical survey of the Levinson method, bringing together Young's streamlined proof for ζ(s) and Wu's twisted-moment treatment for Dirichlet L-functions. However, the manuscript contains no new mathematical result. Its main value lies in assembling known results and numerical constants. The strength of the manuscript is its honest identification of sources: the proof of Levinson's theorem is explicitly attributed to Young [7], and the Dirichlet generalization is attributed to Wu [13]. The stated numerical constants are consistent with those in the cited literature. On the other hand, the exposition is not self-contained: several load-bearing estimates are asserted without proof and explicitly deferred to [13]. In particular, the error-term analysis behind Theorem 3.2 is not carried out in the manuscript, and the quoted Levinson inequality and sieve evaluation in Section III.3 are used as black boxes. Thus the paper cannot serve as a proof of Theorem 3.1 as written, though the underlying theorem is true by the cited work. The significance of the manuscript depends on the venue: as an expository

major comments (3)
  1. [III.2, Theorem 3.2] The proof of the decisive error term is not supplied. After stating "After some long calculation using various other previously known bounds, we prove Z ≪ ..." and "This completes the proof of the proposition," the text immediately adds: "The proof of the error term is quite lengthy and is given in more details in [13]." This is a citation, not a proof. Since Theorem 3.1's constants 0.4172 and 0.4074 are read off from Theorem 3.2 with θ=4/7−ε and the special coefficient shape a(n)=μ(n)(P1+P2·Σ_{p|n,p≤y^{3/4}}P), the claimed lower bound has no support in this manuscript unless the bound on Z and the transition to the stated error term are proved here. Moreover, Theorem 3.2's error term is written as O(R^{1−ε0}); with R=1.3 this is O(1), which is meaningless as an asymptotic error term. Compare the proposition's O(T^{−ε0})—this is very likely a typo for O(T^{1−ε0}), and must be corrected.
  2. [III.3] The final derivation of κ(χ)>0.4172 and κ*(χ)>0.4074 depends on two quoted results that are not proved: the Levinson-type inequality κ(χ) ≥ 1 − (1/R) log(T^{−1} I_R(Q,χ)) + o(1) (and the analogous statement for simple zeros when Q is linear), and the sieve estimate Σ(α,β) = q/φ(q) θ L M + O((log log)^7 log^{−2} y). These are essential to convert the asymptotic of Theorem 3.2 into the stated numerical bounds. The manuscript gives no derivation of either, so the proof of Theorem 3.1 is incomplete at its final step. Either the proofs should be included, or the paper should be explicitly reframed as a survey relying on [13] for these ingredients.
  3. [II.3, Lemma 2.3 and the 'Result'] Even in the Levinson/Young part, the exposition is a sketch rather than a proof. Lemma 2.3 is asserted to follow from a contour manipulation and an unproved 'Result' that supplies the main term of J_{α,β}(M). The statement 'The proof is completed using the following result, qed' is not a proof of that result. Similarly, the assertion after contour shifting that 'the new contour of integration gives O(T^{1−ε})' is not substantiated. If the paper's goal is to present a self-contained proof of Levinson's theorem, these gaps must be filled; if the goal is a survey, the text should say so explicitly.
minor comments (5)
  1. [I.1, Theorem 1.1] The displayed analytic continuation formula is garbled: 'Γ((1−2)/2)' should presumably be 'Γ((1−s)/2)', and the notation around the incomplete gamma factor is unclear. There are also typos ('cosnider', 'its beneficial').
  2. [I.7] The text says 'The non-trivial zeroes of ζ(s) lie in the strip 0≤Re(s)≤1' in the Dirichlet L-function section; this should refer to L(s,χ), not ζ(s).
  3. [II.4] The name 'Balasubramaian' should be 'Balasubramanian'. The displayed asymptotic for the zeta-moment includes the phrase 'where χ is a primitive character', but no character appears in that formula.
  4. [III.2] In the proof of Theorem 3.2, the notation Z is introduced for 'the term in the expression of I_i involving E_i' but the reader is not told what E_i is; the decomposition I_i = M_i + R_i + E_i is only explained by reference to Lemma 3.4. Please spell out the decomposition.
  5. [III.1] The notation e(x) is used (e.g., 'e(−xv/q)') without definition; standard is e(x)=e^{2πix}. Also 'zamd' should be 'z and'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the report is an externally sourced exposition; its load-bearing error term is deferred to Wu [13], an independent published source, not a self-citation or fitted input.

full rationale

The paper is a digest of external results, primarily Young [7] for the zeta-function case and Wu [13] for Dirichlet L-functions. The headline bounds κ(χ)>0.4172 and κ*(χ)>0.4074 in Theorem 3.1 are not derived from data or from the present authors' own prior work; they are outputs of Wu's asymptotic formula (Theorem 3.2) with explicitly stated choices of R, θ, and polynomial coefficients in §III.3. Those choices are optimization parameters within an already-established formula, not parameters fitted to the target lower bound. The single most load-bearing step in the report—the error-term bound in §III.2—is not proved in the manuscript; the text says 'The proof of the error term is quite lengthy and is given in more details in [13].' That is a completeness gap and a reliance on external published work, but it is not circular: the source [13] is not authored by the present paper's authors, and no self-citation chain, no ansatz smuggled in via the authors' own prior work, and no renaming of a fitted quantity as a prediction is present. Per the rules, reliance on an externally falsifiable published theorem does not raise the circularity score. Score 0.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The report contributes no parameter-free derivation of its own: every theorem (Levinson κ≥1/3; Conrey ≥2/5; Wu κ(χ)>0.4172, κ*(χ)>0.4074) is attributed to prior work ([7], [10], [13]), and the numerical constants in Theorem 3.1 depend on polynomial and shift parameters chosen in §III.3 whose optimization is not shown. The load-bearing error term of the Dirichlet twisted-moment asymptotic is explicitly deferred to [13]. Axioms listed cover standard background, the approximate functional equation, an unproved 'Result' in §II.3, the Levinson inequality, the twisted-moment asymptotic itself, and the sieve-theoretic evaluation of Σ. No invented entities (particles, forces, fields) are introduced; this is a zero-free-line counting argument within established analytic number theory.

free parameters (7)
  • R (shift of σ0, κ bound) = 1.3
    Chosen in §III.3 to 'obtain K(χ)>0.4172'; no optimality derivation shown in the report.
  • R (shift of σ0, κ* bound) = 1.116
    Chosen in §III.3 for the simple-critical-zero bound κ*(χ)>0.4074.
  • θ (mollifier length exponent) = 4/7 − ε
    Set in §III.3 to reach Conrey's θ=4/7 range; Theorem 3.2(B) only gives ε0>0 for this θ under the special coefficient shape.
  • Q(x) coefficients (κ bound) = 1−0.642x−1.227(x²/2−x³/3)−5.178(x³/3−x⁴/2+x⁵/5)
    §III.3: 'we choose ... to obtain K(χ)>0.4172'; coefficients imported from Wu's optimization, not derived here.
  • P1,P2,P coefficients (κ bound) = P1=x−0.617x(1−x)−0.125x²(1−x)−0.148x³(1−x); P2=x; P=1.55x−1.564x²+0.177x³
    §III.3; define the mollifier weights; determine the constant M and hence the 0.4172 figure; no derivation of the optimization shown.
  • P1,P2,P coefficients (κ* bound) = P1=x−0.525x(1−x)−0.183x²(1−x)−0.085x³(1−x); P2=x; P=0.838x−0.938x²−0.084x³
    §III.3; paired with Q(x)=1−1.032x to obtain κ*(χ)>0.4074.
  • P, Q, R, θ in the Levinson illustration (§II.3) = P=x, Q=1−x, R=1.3, θ=0.5 → c≈2.35, κ≥0.35
    Numeric evaluation in §II.3; used to reproduce Levinson's 1/3 bound. R and θ are hand-chosen.
assumptions (7)
  • standard math Standard facts on ζ(s) and Dirichlet L-functions: analytic continuation, functional equations, trivial zeros from Γ poles, zero-free region, PNT with error term (Section I, cited to [1–5]).
    Section I lays these out as background. They are standard textbook results used throughout the Levinson machinery (e.g., the functional equations for ξ(s) and ξ(s,χ); the approximate functional equation Theorem 1.3).
  • domain assumption Approximate functional equation for general L-functions (Theorem 1.3, §I.11), cited to Iwaniec–Kowalski [5].
    Used to derive Lemma 2.1 for ζζ and the analogous start of the Dirichlet moment in §III.1. Restated without proof in the report.
  • domain assumption 'Result' (§II.3): J_{α,β}(M) = (log M)^{i+j−1}/(i!j!) · d²/dxdy M^{αx+βy} ∫₀¹ (x+u)^i (y+u)^j du |_{x=y=0} + O(L^{i+j−2}).
    Stated as 'Result' with no proof; it completes the proof of Lemma 2.3 and is the decisive evaluation in Young's proof of the Levinson constant. The report supplies no derivation.
  • domain assumption Levinson inequality for Dirichlet L-functions: κ(χ) ≥ 1 − (1/R) log(T^{−1} I_R(Q,χ)) + o(1) for suitable R>0, and the same expression bounds the simple critical zeros when Q is linear (§III.3).
    Quoted as 'Levinson found a generalized inequality' without proof. It is the bridge converting the mollified second-moment asymptotic (Theorem 3.2) into the proportion bounds in Theorem 3.1.
  • domain assumption Twisted-moment asymptotic, Theorem 3.2 and the Proposition (§III): g(α,β,w) = main term + O(T^{−ε0}), with ε0>0 for θ<17/33 and for θ<4/7 under the special coefficient shape (B).
    This is the engine of the Dirichlet result. The report proves the main term's outline (lemmas 3.1–3.5) but the error term is not derived: §III.2 states a bound Z ≪ y^{7/8}T^{−1/2+11η/2+ε} + y^{7/4}T^{−1+11η/2+ε} 'after some long calculation' and then defers the reader to [13].
  • domain assumption Sieve-theoretic asymptotic Σ(α,β) = q/φ(q) · θ · Lχ · M + O((log log y)^7 log^{−2} y) (§III.3).
    Quoted 'after some calculations using the lemmas... and some previously known bounds from Sieve theory.' No derivation shown; the value of M depends on the choice of P1, P2, P, and through it on the final constants 0.4172/0.4074.
  • domain assumption Conditions: log q = o(log T), q any positive integer, a(m) ≪ m^ε, y = T^θ (Theorem 3.1/3.2).
    Regime assumptions stated in Theorem 3.1 ('for sufficiently large T with log q = o(log T)') and Theorem 3.2; they delimit where the asymptotic is claimed to hold.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Levinson's theorem and its generalization for Dirichlet L-functions." pith.science (2026). https://pith.science/paper/ZBFBIAKP

@misc{pith2026251106109,
  author       = {Pith},
  title        = {Pith review of: Levinson's theorem and its generalization for Dirichlet L-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZBFBIAKP}},
  note         = {Machine review of arXiv:2511.06109}
}
read the original abstract

In this report, we present a proof of Levinson's theorem, following the ideas of Matthew P. Young in 2010, which states that one-third of the non-trivial zeros of the Riemann zeta function lie on the critical line, i.e. the line Re(s) = 1/2, using a mollified second moment of the zeta-function. Later, we present a generalized result for Dirichlet L-functions by Xiaosheng Wu in 2018, using Levinson's method, showing that more than two-fifths of the non-trivial zeros of Dirichlet L-functions are on the critical line. Moreover, more than two-fifths of the non-trivial zeros are simple and on the critical line, using a longer mollifier than in Levinson's original proof. This generalizes a result by Conrey from 1989 that the Riemann zeta-function has at least two-fifths of its zeros on the critical line.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 1 linked inside Pith

  1. [7]

    Young.A Short Proof of Levinson ’s Theorem

    Matthew P. Young.A Short Proof of Levinson ’s Theorem. arXiv:1002.4403v1, 2010

    arXiv 2010

  2. [10]

    J. B. Conrey.More than two-fifths of the zeroes of the Riemann zeta-function are on the critical line. J. Reine Angew. Math, 1989

    1989

  3. [13]

    Math- ematische Zeitschrift, 2018

    Xiaosheng Wu.The twisted mean square and critical zeroes of Dirichlet L-functions. Math- ematische Zeitschrift, 2018

    2018

  4. [1]

    Apostol.Introduction to Analytic Number Theory

    Tom M. Apostol.Introduction to Analytic Number Theory. Springer-Verlag, 1976

    1976

  5. [2]

    Montgomery and Robert C

    Hugh L. Montgomery and Robert C. Vaughan.Multiplicative Number Theory: I. Classical Theory. Cambridge University Press, 2006

    2006

  6. [3]

    Springer-Verlag, 1980

    Harold Davenport.Multiplicative Number Theory. Springer-Verlag, 1980

    1980

  7. [4]

    Murty.Problems in Analytic Number Theory

    Ram M. Murty.Problems in Analytic Number Theory. Springer, 2001

    2001

  8. [5]

    American Mathemat- ical Society, 2004

    Henryk Iwaniec and Emmanuel Kowalski.Analytic Number Theory. American Mathemat- ical Society, 2004

    2004

Show all 15 references
  1. [6]

    Brian Conrey.Riemann ’s Hypothesis

  2. [8]

    Bettin and S

    S. Bettin and S. Gonek.Theθ=∞conjecture implies the Riemann hypothesis. Mathe- matika, 2017

    2017

  3. [9]

    Balasubramanian, J

    R. Balasubramanian, J. B. Conrey, and D. R. Heath-Brown.Asymptotic mean-square of the product of the Riemann zeta function and a Dirichlet Polynomial. J. Reine Angew. Math, 1985. 26

    1985

  4. [11]

    Bettin, V

    S. Bettin, V. Chandee, and M. Radziwill.The mean square of the product of the Riemann zeta-function with Dirichlet polynomials. J. Reine Angew. Math, 2017

    2017

  5. [12]

    Pratt and N

    K. Pratt and N. Robles.Perturbed moments and a longer mollifier for critical zeroes ofζ. Res. Number Theory, 2018

    2018

  6. [14]

    Conrey, H

    J. Conrey, H. Iwaniec, and K. Soundararajan.Critical zeroes of Dirichlet L-functions. J. Reine Angew. Math, 2013

    2013

  7. [15]

    Xiaosheng Wu.Distinct zeroes and simple zeroes for the family of Dirichlet L-functions. Quart. J. Math, 2016. 27

    2016

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.