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REVIEW 5 major objections 5 minor 36 references

Beweis der Riemannschen Vermutung \"uber ein reguliertes normiertes Integralmodell

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims to prove the Riemann Hypothesis by constructing a weighted area integral $W(R)$ whose convergence is equivalent to every nontrivial zero lying on the critical line $\Re(s)=1/2$.

desk verdict A serious but invalid RH proof attempt: both directions of the claimed equivalence fail because the excised zero disks are quietly reinserted in the estimates. read the letter →

arxiv 2505.23238 v2 pith:XUMWBAUP submitted 2025-05-29 math.GM

Dennis-Magnus Welz This is my paper
classification math.GM MSC 11M2611M06
keywords RiemannhypothesiszetazeroscriticallineareaintegralregulatedanalyticcriterionDirichletseriesMellinrepresentation
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 prove the Riemann Hypothesis by tying it to the convergence of a weighted area integral over the critical strip. The integral $W(R)$ averages $|\zeta(s)|^{-\lambda}|\Re(s)-\tfrac12|^{-p}$ over a strip from which small disks around every zero and around the pole at $s=1$ have been removed. The central claim is that $W(R)$ has a finite limit as $R\to\infty$ if and only if every nontrivial zero of $\zeta$ lies on the critical line $\Re(s)=\tfrac12$. Having argued that any off-line zero forces divergence and that $W(R)$ actually converges, the paper concludes that the Riemann Hypothesis is true. If the argument is right, RH would follow from a single analytic convergence statement rather than from numerical or spectral input.

What carries the argument

The central object is the regulated normalized area integral $W(R)=\frac{1}{2R}\int\int_{D_R^{\mathrm{reg}}} |\zeta(s)|^{-\lambda}|\Re(s)-\tfrac12|^{-p}\,dA(s)$. The domain $D_R^{\mathrm{reg}}$ is the vertical strip $\{0\le\Re(s)\le1,\,|\Im(s)|\le R\}$ with disks $B(\rho_n,\varepsilon_n)$ around each nontrivial zero and a disk around the pole $s=1$ removed; the excision radii are $\varepsilon_n=(n+N_0)^{-\alpha}$ with $\alpha>1$, chosen so the total removed area is finite. The exponents $\lambda\ge2$ and $0<p<1$ make $|\zeta|^{-\lambda}$ dominate the horizontal weight $|\Re(s)-\tfrac12|^{-p}$, so a zero off the critical line becomes a potential divergence source. The proof then runs on two halves: a case analysis showing that any off-line zero makes $W(R)$ diverge, and a convergence proof for $W(R)$ built on exact representations of $\zeta$, whose combination forces all zeros onto the critical line.

What would settle it

Build a model function with exactly one simple zero $\rho$ off the critical line and integrate the same regulated integrand over the same strip, removing a disk of radius $\varepsilon_n=(n+N_0)^{-\alpha}$ around $\rho$ with the paper's parameters $\lambda=2$, $p=\tfrac12$. The local annulus contribution is proportional to $\int_{\varepsilon_n}^{\varepsilon} r^{1-\lambda}\,dr$, which is finite for every $\varepsilon_n>0$, so a finite limit, or a divergence rate different from the paper's prediction, would directly contradict the claimed equivalence.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is an equivalence: $\lim_{R\to\infty} W(R)<\infty$ holds exactly when all nontrivial zeros of $\zeta(s)$ satisfy $\Re(s)=\tfrac12$. The construction is deliberately singularity-sensitive: near a zero $\rho$ of multiplicity $m$, $|\zeta(s)|$ behaves like $|s-\rho|^m$, so the integrand behaves like $|s-\rho|^{-\lambda m}$ times a horizontal weight, and with $\lambda\ge2$ this is non-integrable at an off-line zero if the integration actually reaches the zero. Chapter 4 classifies all zero configurations and asserts that every deviation from the critical line therefore forces divergence; Chapter 5 claims to prove convergence of $W(R)$ using only the alternating Dirichlet series and a regulated Mellin representation, without knowing zero locations. The formal conclusion in Section 6.4 is that the Riemann Hypothesis is true.

Load-bearing premise

The proof that one off-line zero makes the integral diverge assumes the integral is taken down to that zero, even though the regulated domain removes a positive-radius disk around every zero; without a separate argument that the removed disks still accumulate divergence, the necessity direction is not established.

Editorial extensions

If this is right

  • If the central claim is correct, all nontrivial zeros of $\zeta(s)$ lie on $\Re(s)=\tfrac12$, so the Riemann Hypothesis follows as a corollary.
  • The equivalence turns RH into a single analytic convergence check, with no need to locate individual zeros; the excision radii depend only on the zero index.
  • The method is claimed to transfer to other zeta and $L$-functions, giving analogous convergence criteria for their zero distributions.
  • Because the normalization by $1/(2R)$ cancels the strip's linear area growth, a finite limit detects structural zero misplacement rather than the trivial growth of the domain.
  • Any hypothetical off-line zero would force $W(R)$ to diverge, so the criterion would expose the smallest deviation from the critical line.

Reading between the lines

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

  • A natural stress test is to apply the same construction to a meromorphic function with one prescribed off-line zero and summable excision radii; if the regulated limit stays finite, the necessity direction would need additional hypotheses not stated in the paper.
  • If the convergence proof in Chapter 5 genuinely uses only exact representations of $\zeta$, the same integrand could provide a concrete analytic criterion for Dirichlet $L$-functions or other $L$-functions with the same zero symmetries, a transfer the paper only sketches.
  • The vertical projection $\Phi(t)$ introduced in Chapter 7 suggests a numerical probe: tracking whether $W(R)$ stabilizes for large $R$ in high-precision computation would be consistent with the paper's conclusion, although it would not by itself prove the limit statement.
  • The paper's equivalence, if correct, would place RH in the family of integral-convergence criteria, potentially connecting to existing zero-counting and explicit-formula methods without requiring the zeros to be known in advance.
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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

5 major / 5 minor

Summary. The manuscript claims to prove the Riemann Hypothesis by constructing a regulated, normalized surface integral W(R) over a vertical strip in the critical strip, with integrand JC(s) = 1/(|zeta(s)|^lambda |Re(s)-1/2|^p), and by excising small disks around the zeros and the pole. The central theorem is stated as lim_{R->infty} W(R) < infinity if and only if the Riemann Hypothesis holds. The proof proceeds in two directions: a necessity direction claiming that any zero off the critical line makes the integral diverge, and a sufficiency direction claiming that W(R) converges without assuming RH, using the alternating Dirichlet series or a regulated Mellin representation. The paper concludes that both directions imply the Riemann Hypothesis.

Significance. If the central equivalence were correct, this would be a major breakthrough in analytic number theory, reducing RH to a convergence criterion for an explicit integral and avoiding all known difficult aspects of the problem. The manuscript is clearly organized, states definitions and lemmas in a formal style, and references standard works. However, the claimed proof rests on load-bearing internal inconsistencies: the necessity direction integrates over disks that the construction explicitly removes, and the convergence proof uses a lower bound m_R that depends on R and degenerates as R grows. These are not merely presentation issues; they invalidate both directions of the proposed equivalence. The paper also contains a later section that fits constants to known zeros, which is empirical rather than a derivation. For these reasons, the significance of the claimed result cannot be recognized in the current form.

major comments (5)
  1. [§5.4.2, Lemma 5.4.1 and Satz 5.4.2] The divergence claim for an off-line zero rho computes the local contribution I_rho(R) as (1/(2R)) times the integral of r^{1-lambda m} from r = 0 to epsilon, i.e., over the full disk B(rho, epsilon). This contradicts Definition 3.5.3, which removes the disk B(rho_n, epsilon_n) from D_reg^R, with epsilon_n = (n+N0)^{-alpha}. For the actual domain the radial integral starts at r = epsilon_n and equals (epsilon^{2-lambda m} - epsilon_n^{2-lambda m})/(2-lambda m), which is finite for each fixed R and tends to 0 after division by 2R. Hence the inequality W(R) >= I_rho(R) = infinity is false, and the necessity direction of Satz 6.2.1 has no basis.
  2. [§4.3.2, Lemma 4.3.2] The convergence proof bounds W(R) by m_R^{-lambda} times the finite integral of |sigma - 1/2|^{-p}, and then claims that all estimates are independent of R. This is incorrect because m_R = min_{s in D_reg^R} |zeta(s)| depends on R: as R increases, more zeros enter the strip and the excision radii epsilon_n tend to 0, so m_R tends to 0. The factor m_R^{-lambda} therefore grows without bound, and the presented estimate does not imply limsup_{R->infty} W(R) < infinity. The same defect appears in §5.4.3, where the analogous bound is used.
  3. [§5.2 vs §6.4.1] In the claimed proof that W(R) -> 0 under RH, the contribution from S2(R) is bounded by (1/(2R)) * C R^{alpha p} * Area(S2(R)) <= (1/(2R)) * C R^{alpha p} * 4R = 2 C R^{alpha p}, which diverges for any alpha > 1 and p > 0. The text states that alpha is chosen so that the term is controlled, but no such choice exists because alpha p > 0. Moreover, the claim that JC is bounded by a constant M on S1(R) is not uniform in R, since near the boundaries of the excised disks |zeta|^{-lambda} grows like epsilon_n^{-lambda m}. Thus the convergence statement for the RH case is not proved.
  4. [§7.1.5] Theorem 5.2.1 states the equivalence with lim_{R->infty} W(R) = 0, while the abstract and Satz 6.4.1 use lim_{R->infty} W(R) < infinity. These are different statements: the proof of Lemma 4.3.2 (if correct) would give convergence to 0 under RH, whereas Section 5.4 claims only finiteness. The manuscript does not reconcile these formulations, and the formal main theorem is therefore ambiguous.
  5. [§5.4.2] The 'asymptotic approximation' for the zero ordinates gamma_n contains constants a, b, c, d that are numerically fitted to the known zeros of the zeta function. This is empirical curve fitting, not a derivation from the integral model, and it does not provide independent support for the main theorem. In addition, the vertical projection Phi(t) in §7.1.1 is introduced under the assumption of RH, so the reconstruction argument is conditional on the statement being proved.
minor comments (5)
  1. [§4.3.1] In Satz 3.6.3 the local integrals are computed over r from epsilon to delta and are finite, but the concluding sentence says that the local integrals around the zeros diverge 'grundsätzlich'. This contradicts the preceding calculation; the divergence is removed by the excision, and the exposition should be corrected to avoid confusion.
  2. [Definition 3.5.1] The summary of the case distinction says 'konvergiert exakt dann gegen null', while the stated criterion is lim W(R) < infinity. These are not the same, and the inconsistency should be resolved.
  3. [§4.1] The symbol delta is used both for the half-width of the vertical strip in Definition 3.5.1 and for the deviation beta - 1/2 in Lemma 4.3.3. This overloaded notation makes the local asymptotic formulas harder to follow.
  4. [§7.1.5] There is a typo 'Definitoin' in the sentence introducing epsilon_n, and the phrase 'alle singulären Beiträge ... vollständig reguliert' is unclear because the excision is a regularization of the domain, not a deletion of the contributions.
  5. [Bibliography] The approximation formula has a typo in the phrase 'für kleineren' and the displayed formula contains an undefined parameter 'n' and a term 'd/n^7' whose origin is not explained. The figures referenced as Abbildung 7.1 and 7.2 are not included in the manuscript text.

Circularity Check

2 steps flagged · score 7.0 of 10

The off-line divergence is computed over the disk that Definition 3.5.3 removes; the necessity direction of the RH equivalence is an artifact of the construction, and Section 7.1.5 fits its constants to the zeros it claims to predict.

  1. self definitional [Definition 3.5.3/3.5.4, Lemma 4.3.3 (and Lemmas 4.3.4–4.3.8)]
    "Definition 3.5.3: Dreg_R := DR \ (⋃_{ρn∈DR} Kn ∪ Kpol). Lemma 4.3.3: Iρ(R) := 1/2R ∫∫_{B(ρ,ε)} JC(s)dA(s) ≥ 1/2R · 2π/|δ|^p ∫_0^ε r^{1−λm}dr = π/(R·|δ|^p) ∫_0^ε r^{1−λm}dr. ... Somit ist Iρ(R)=∞ und damit W(R)≥Iρ(R)→∞ für R→∞."

    W(R) is defined over Dreg_R, from which every disk K_n = B(ρ_n, ε_n) around a zero is removed. Lemma 4.3.3 lower-bounds W(R) by I_ρ(R), the integral over B(ρ,ε) — exactly the excised neighbourhood. The divergence comes from ∫_0^ε r^{1−λm}dr over the removed singularity. On the actual domain Dreg_R, the radial integration starts at r = ε_n, so the local contribution is finite for each fixed R. Hence W(R) ≥ I_ρ(R) = ∞ is false, and the necessity direction of Satz 6.2.1/6.4.1 has no basis; the claimed sensitivity is re-inserted by construction.

  2. fitted input called prediction [Section 7.1.5, Asymptotische Näherung für γn]
    "Durch Umkehrung der Riemann-von Mangoldt-Zählfunktion erhält man die asymptotische Formel γn≈ 2πn/log n ( 1 + a/√(n log n) + b/n + c log log n/log n + d/n^7 ), wobei die Konstanten a,b,c,d numerisch aus der Auswertung der tatsächlichen Nullstellen der Zetafunktion bestimmt wurden."

    The section presents this as a 'Streng analytische Herleitung der asymptotischen Nullstellennäherungsformel' from the integral model and Fourier transform. But the constants a,b,c,d are explicitly fitted to the true zero ordinates. The good agreement with computed zeros is therefore a measure of the fit, not an independent prediction from W(R); the empirical zeros are inputs to the formula, not outputs of the derivation.

full rationale

The paper is not saved by a self-citation chain: it cites only standard references and no self-citations are load-bearing. The main circularity is internal. Definition 3.5.3/3.5.4 defines W(R) on Dreg_R with small disks around every zero removed. Lemma 4.3.3 obtains divergence from an off-line zero by integrating over the full disk B(ρ,ε), i.e. by treating the excised singularity as part of the domain; the inequality W(R) ≥ I_ρ(R) is invalid. Lemmas 4.3.4–4.3.8 inherit this same cut-off confusion. Consequently Satz 6.2.1 (RH necessary for convergence) and the final equivalence are unsupported. Separately, Section 5.4's convergence proof claims the estimates are independent of R although the lower bound m_R depends on R and can shrink; this is a further correctness gap, but it is not itself a circular step. Section 7.1.5 is a clear fitted-input/prediction case: constants are numerically determined from the actual zeros and then the formula is presented as an analytic derivation. The circularity score is 7 because the central necessity direction of the proof reduces to a construction artifact, while the fitted constants concern a secondary numerical claim.

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

The central derivation depends on several unspecified exponents (lambda, p, alpha, N0, delta) and on the fitted constants a, b, c, d of Chapter 7. It also relies on two unproved or false premises: that an excised zero neighborhood still contributes a divergent local integral, and that the minimum m_R can be treated as a uniform constant. The only genuinely standard input is the classical analytic machinery of zeta(s).

free parameters (6)
  • lambda = lambda >= 2 (unspecified)
    Exponent in JC(s) = |zeta(s)|^{-lambda} |Re(s)-1/2|^{-p}; chosen to make zero singularities nonintegrable (lambda m >= 2 for every zero multiplicity m). No specific value is fixed.
  • p = 0 < p < 1 (unspecified)
    Exponent controlling the weight near the critical line; p < 1 makes |x|^{-p} locally integrable along the line. Used in all convergence claims.
  • alpha = alpha > 1 (unspecified)
    Exponent in the excision radii epsilon_n = (n + N0)^{-alpha}; controls the sum of removed areas and affects the size of m_R, but the paper treats the choice as arbitrary.
  • N0 = real constant, unspecified
    Shift parameter in epsilon_n; no value is given.
  • delta = 0 < delta < 1/2 (unspecified)
    Half-width parameter used in the strip [delta, 1-delta] in the convergence estimates of Section 5.4.
  • a, b, c, d = not stated in text
    Constants in the asymptotic formula for gamma_n in Section 7.1.5; the paper says they were numerically determined from the actual zeros, which is fitting to data.
assumptions (6)
  • standard math Analytic continuation and functional equation of zeta(s)
    Used throughout Chapter 2 to locate zeros and establish symmetries; a standard result.
  • standard math Zeros of zeta are isolated and discrete
    Used to excise disks B(rho_n, epsilon_n) and to claim a finite zero count on compact strips (Satz 2.2.6).
  • standard math Riemann-von Mangoldt counting formula N(T)
    Used in Section 2.2.7 and again in Section 7.1.4 to approximate gamma_n.
  • ad hoc to paper A zero's local contribution to W(R) is computed over the full disk including the excised neighborhood
    Lemma 4.3.3 integrates r from 0 to epsilon over B(rho, epsilon), although Definition 3.5.3 removes B(rho, epsilon_n) from Dreg_R. This is the load-bearing premise of the necessity direction.
  • ad hoc to paper The lower bound m_R = min_{Dreg_R} |zeta| can be used as a constant independent of R
    Section 5.4.2 claims the estimates are independent of R, but m_R decreases like (R log R)^{-alpha} as R grows because epsilon_n decays. The claimed uniform convergence bound is unsupported.
  • domain assumption Equivalence N_Phi(T) approx N(T) between the frequency-counting function and the Riemann-von Mangoldt count
    Section 7.1.4 assumes this equivalence to reconstruct zero ordinates from the Fourier structure; no proof is supplied.
invented entities (1)
  • Regulated normalized surface integral W(R) with integrand JC(s) = |zeta(s)|^{-lambda} |Re(s)-1/2|^{-p}
    purpose: Central criterion claimed to converge exactly under RH; used to infer RH from a convergence proof.
    The object is constructed by excising small disks around every zero, so its claimed sensitivity to off-line zeros is not independently evidenced; the divergence argument reintroduces exactly the excised disks.

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Pith. "Pith review of Beweis der Riemannschen Vermutung \"uber ein reguliertes normiertes Integralmodell." pith.science (2026). https://pith.science/paper/XUMWBAUP

@misc{pith2026250523238,
  author       = {Pith},
  title        = {Pith review of: Beweis der Riemannschen Vermutung \"uber ein reguliertes normiertes Integralmodell},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XUMWBAUP}},
  note         = {Machine review of arXiv:2505.23238}
}
read the original abstract

We prove the Riemann Hypothesis via an analytically regulated surface integral over the critical strip of the Riemann zeta function. The key idea is that the convergence of this normalized integral is equivalent to the condition that all non-trivial zeros lie on the critical line. By constructing a singularity-sensitive integrand and removing infinitesimal disks around the poles, we isolate all divergence contributions analytically. The resulting integral model is independent of specific zero locations and provides a purely analytic criterion equivalent to the Riemann Hypothesis.

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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    eine vollständige Fallunterscheidung sämtlicher geometrisch möglicher Null- stellenkonfigurationen, die belegt, dass jede Abweichung von RH zur Diver- genz führt,

    work page

  2. [2]

    Die Methode istmethodisch unabhängigvon spektralen, statistischen oder nume- rischen Ansätzen und benötigt keine Kenntnis der exakten Nullstellen

    ein analytischer Nachweis der tatsächlichen Konvergenz vonW (R) unter ausschließlicher Verwendung exakter Darstellungen vonζ(s) (alternierende Dirichletreihe, regulierte Mellin-Darstellung). Die Methode istmethodisch unabhängigvon spektralen, statistischen oder nume- rischen Ansätzen und benötigt keine Kenntnis der exakten Nullstellen. Darüber hinaus ist ...

    work page

  3. [3]

    Da das Flächenmaß in einer Umgebung vonL einer Produktmenge entspricht, gilt auch im komplexen Raum die lokale Integrabilität

    LokaleL1-Integrabilität Wegen 0<p< 1 ist die Funktionx↦→|x|−p lokal integrierbar in einer Umgebung umx = 0 bezüglich des eindimensionalen Maßes. Da das Flächenmaß in einer Umgebung vonL einer Produktmenge entspricht, gilt auch im komplexen Raum die lokale Integrabilität. Außerdem ist|ζ(s)|−λ aufDreg R beschränkt, da Nullstellenbereiche ausgeschnitten sind...

    work page

  4. [4]

    September 2025 arXiv:2505.23238v2 [math.GM] 7 Aug 2025 Mathematische Beweisarbeit zur vollständigen Charakterisierung der Nullstellen der Riemannschen Zetafunktion Eingereicht zur Begutachtung an der Technischen Hochschule Ulm Dennis-Magnus Welz dennismagnuswelz@gmail.com i Zusammenfassung Diese Arbeit entwickelt und beweist ein neuartiges, vollständig an...

    work page Pith review arXiv 2025

  5. [5]

    a complete case distinction of all geometrically possible zero configurations, proving that any violation of RH leads to divergence,

    work page

  6. [6]

    The method ismethodologically independentof spectral, statistical, or numerical approaches and does not require knowledge of the exact zeros

    an analytical proof of the actual convergence ofW (R) using only exact representations ofζ(s) (alternating Dirichlet series, regulated Mellin repre- sentation). The method ismethodologically independentof spectral, statistical, or numerical approaches and does not require knowledge of the exact zeros. Moreover, the criterion can be transferred to other ze...

    work page

  7. [7]

    Notwendigkeit: Eine vollständige Fallunterscheidung aller geometrisch möglichen Nullstellenkonfigurationen zeigt, dass jede Abweichung von der kritischen Linie – sei sie noch so gering – zu einer Divergenz des Integrals 1 Kapitel 1: Einleitung und Motivation im GrenzfallR→∞ führt

    work page

  8. [8]

    Tatsächliche Konvergenz:Es wirdohne jede Vorannahme über die La- ge der Nullstellen streng analytisch nachgewiesen, dass W (R) im Limes R→∞ endlich bleibt. Dies geschieht ausschließlich unter Verwendung ex- akter Darstellungen der Zetafunktion (alternierende Dirichletreihe, regulier- te Mellin-Darstellung) und unter vollständiger Kontrolle aller auftreten...

    work page 1914

Show all 36 references
  1. [9]

    Stetigkeit aufDreg R \L Abgesehen von den ausgeschlossenen Kreisscheiben B(ρn,εn) um die Nullstellenρn sowie der Menge L := { s∈D reg R | Re(s) = 1 2 } , ist JC(s) das Produkt zweier stetiger Funktionen, daζ(s) holomorph und nicht verschwindend aufDreg R \ ⋃ B(ρn,εn) ist und R...

  2. [10]

    Verhalten an der kritischen LinieL Die MengeL ist eine eindimensionale Linie im zweidimensionalen Flächenmaß und besitzt somit Maß Null (vgl. [4, Kap. 1, S. 3–4]). Punktweise Unstetigkeit oder Divergenz aufL beeinträchtigt das Lebesgue-Integral nicht

  3. [11]

    RH-Fall: Alle nicht-trivialen Nullstellen liegen exakt auf der kritischen Linie Re(s) = 1 2

  4. [12]

    • Symmetrische Viererpaarbildung von Nullstellen

    Abweichungen vom RH: • Einzelne Nullstelle(n) abseits der kritischen LinieRe(s)̸= 1 2. • Symmetrische Viererpaarbildung von Nullstellen. • Asymmetrische, zentrisch oder nicht-zentrisch symmetrische Paare. • Nullstellen nahe am Rand des kritischen StreifensRe(s) = 0 oder 1. • M...

  5. [13]

    Betrachte die vier zugehörigen Punkte im kritischen Streifen: β±iγ und (1−β)±iγ

    Durch die funk- tionale Gleichung und komplexe Konjugation der Zetafunktion ergibt sich auto- matisch das symmetrische Viererpaar {ρ, ¯ρ, 1−ρ, 1− ¯ρ}. Betrachte die vier zugehörigen Punkte im kritischen Streifen: β±iγ und (1−β)±iγ. Da β ̸= 1 2, liegt jede dieser vier Nullstell...

  6. [14]

    Diese Nullstellen liegen somit **nicht** auf der kritischen Linie und erfüllenβ̸= 1 2

    oder β∈ ( 1 2, 1), wobeiβ sehr nahe an einem Randwert0 oder 1 liegt. Diese Nullstellen liegen somit **nicht** auf der kritischen Linie und erfüllenβ̸= 1 2. Wie in Lemma 4.3.3 gezeigt, ist der Integrand JC(s) = |ζ(s)|−λ | Re(s)− 1 2|p in einer Umgebung vonρ asymptotisch JC(s)∼ ...

  7. [15]

    diealternierendeDirichlet-Reihe (vollständigeanalytischeFortsetzung)(Satz 2.1.8),

  8. [16]

    Diese Repräsentationen besitzen unterschiedliche analytische Eigenschaften und Gültigkeitsbereiche, sind aber sämtlich geeignet, um das IntegralmodellW(R) konkret auszuwerten

    unddie Mellin-Darstellung(basierendaufderEuler-Maclaurin-Entwicklung)(Satz 2.1.9). Diese Repräsentationen besitzen unterschiedliche analytische Eigenschaften und Gültigkeitsbereiche, sind aber sämtlich geeignet, um das IntegralmodellW(R) konkret auszuwerten. Sie erlauben es un...

  9. [17]

    Wir definieren wie zuvor: Dreg R :={s =σ +it∈ C|σ∈ [δ, 1−δ], t∈ [−R,R ]}\ N(R)⋃ k=1 B(ρk,εk)

    Regulierung des Bereichs. Wir definieren wie zuvor: Dreg R :={s =σ +it∈ C|σ∈ [δ, 1−δ], t∈ [−R,R ]}\ N(R)⋃ k=1 B(ρk,εk). Die Kreisscheiben B(ρk,εk) entfernen alle Punkte, an denen|ζ(s)| < ε, d.h. potenzielle Nullstellen. Diese sind endlich zahlreich im kompakten Streifen

  10. [18]

    Lemma 5.4.1 (Untere Schranke)

    Existenz einer positiven unteren Schranke. Lemma 5.4.1 (Untere Schranke). Für jeden festenR > 0 existiert eine Kon- 37 Kapitel 5: Beweis der Riemannschen Vermutung über das regulierte Integralmodell stante mR > 0, sodass |ζ(s)|≥ mR für alles∈D reg R . Beweis. Da ζ(s) holomorph...

  11. [19]

    Da|ζ(s)|≥ mR, folgt für den Integran- den: JC(s) = 1 |ζ(s)|λ·|σ− 1 2|p≤ 1 mλ R · 1 |σ− 1 2|p

    Abschätzung des Integranden. Da|ζ(s)|≥ mR, folgt für den Integran- den: JC(s) = 1 |ζ(s)|λ·|σ− 1 2|p≤ 1 mλ R · 1 |σ− 1 2|p. Der Ausdruck|σ− 1 2|−p ist über [δ, 1−δ] integrierbar, daδ >0 und p< 1

  12. [20]

    Die Fläche des Rechtecks ist Fläche(Dreg R )≤ (1− 2δ)· 2R

    Integrationsfläche. Die Fläche des Rechtecks ist Fläche(Dreg R )≤ (1− 2δ)· 2R

  13. [21]

    W(R)≤ 1 2R· 1 mλ R · ∫R −R dt· ∫1−δ δ 1 |σ− 1 2|pdσ = 1 mλ R · ∫1−δ δ 1 |σ− 1 2|pdσ

    Abschätzung des Gesamtintegrals. W(R)≤ 1 2R· 1 mλ R · ∫R −R dt· ∫1−δ δ 1 |σ− 1 2|pdσ = 1 mλ R · ∫1−δ δ 1 |σ− 1 2|pdσ

  14. [22]

    Wir betrachten den kritischen Teilbereich des In- tegrationsstreifens entlang der reellen Achse, wobei nur die Nähe zur kritischen Linie σ = 1 2 problematisch ist

    Endliche Konvergenz. Wir betrachten den kritischen Teilbereich des In- tegrationsstreifens entlang der reellen Achse, wobei nur die Nähe zur kritischen Linie σ = 1 2 problematisch ist. Für einen festen kleinen Parameterδ∈ (0, 1

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    betrachten wir das Integral: ∫1−δ δ 1 |σ− 1 2|pdσ. Da die Funktion f(σ) = 1 |σ− 1 2|p eine isolierte Singularität bei σ = 1 2 besitzt, untersuchen wir die Konvergenz des Integrals durch Aufteilung: ∫1−δ δ 1 |σ− 1 2|pdσ = ∫1 2 δ 1 ( 1 2−σ)pdσ + ∫1−δ 1 2 1 (σ− 1 2)pdσ. Dies ist ...

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    Abschätzung des Kerns. Für s = σ +it mit σ∈ [δ, 1−δ], δ > 0, und |t|≤ R fest, betrachten wir getrennt: - Im Bereichx∈ (0, 1): Der Ausdruck ( 1 ex− 1− 1 x ) ist glatt und endlich, da fürx→ 0 1 ex− 1− 1 x =−1 2 +O(x), x → 0. Also gilt: ⏐⏐⏐⏐ ∫1 0 ( 1 ex− 1− 1 x ) xσ−1dx ⏐⏐⏐⏐≤C1(δ...

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    Die Funktion|Γ(s)| ist aufσ∈ [δ, 1−δ], |t|≤ R, holomorph und dort nach unten beschränkt: |Γ(s)|≥ γR > 0 ⇒ |ζ(s)|≤ C γR =:MR

    Schranke der Gammafunktion. Die Funktion|Γ(s)| ist aufσ∈ [δ, 1−δ], |t|≤ R, holomorph und dort nach unten beschränkt: |Γ(s)|≥ γR > 0 ⇒ |ζ(s)|≤ C γR =:MR

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    Regulierung um Nullstellen. Wie in den vorherigen Abschnitten definie- ren wir: Dreg R :={s∈ C| σ∈ [δ, 1−δ],|t|≤ R}\ N(R)⋃ k=1 B(ρk,εk), wobei dieB(ρk,εk) Kreisscheiben um bekannte oder numerisch auffällige Null- stellen sind. Auf diesem Gebiet ist|ζ(s)|≥ mR > 0. 40 Kapitel 5:...

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    AufDreg R gilt: JC(s) = 1 |ζ(s)|λ·|σ− 1 2|p≤ 1 mλ R · 1 |σ− 1 2|p

    Abschätzung des Integranden. AufDreg R gilt: JC(s) = 1 |ζ(s)|λ·|σ− 1 2|p≤ 1 mλ R · 1 |σ− 1 2|p

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    Die Fläche des Streifens beträgt höchstens2R· (1− 2δ), und der Gewichtsfaktor ist fürp< 1 integrierbar: ∫1−δ δ 1 |σ− 1 2|pdσ <∞

    Kontrolle des Integrals. Die Fläche des Streifens beträgt höchstens2R· (1− 2δ), und der Gewichtsfaktor ist fürp< 1 integrierbar: ∫1−δ δ 1 |σ− 1 2|pdσ <∞. Somit: W(R)≤ 1 2R· 2R· 1 mλ R · ∫1−δ δ 1 |σ− 1 2|pdσ <∞. (vgl. Abschnitt 5.4, Paragraph 5.4)

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    Da die Abschätzung unabhängig vonR ist, folgt: lim R→∞ W(R)<∞

    Limes. Da die Abschätzung unabhängig vonR ist, folgt: lim R→∞ W(R)<∞. Satz 5.4.3 (Konvergenz über die regulierte Mellin-Darstellung). Die regulier- te IntegralformW(R) konvergiert für R→∞ , wenn ζ(s) durch die regulierte Mellin-Darstellung eingesetzt wird, und die Nullstellen ...

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