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REVIEW 5 major objections 5 minor 14 references

Holder continuity of an alternating Erdos series on prime K-tuples

T0 review · 5 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read With a damped integral against the Chebyshev function, the paper argues that the alternating Erdős series ∑ (−1)^n n/p_n converges conditionally under the Riemann Hypothesis.

desk verdict Desk-reject; the central identity is false and the paper actually analyzes a damped prime-power sum, not the Erdős series. read the letter →

arxiv 2505.06242 v1 pith:K3QOKTDZ submitted 2025-04-27 math.GM

Nikos Mantzakouras This is my paper · ORCID
classification math.GM MSC 11N0540A05
keywords alternatingErdősseriesconditionalconvergenceRiemann-StieltjesintegralChebyshevfunctionHöldercontinuityRiemannhypothesisprimek-tuplesconjectureYoungcriterion
topics The Riemann HypothesisThe Twin Prime Conjecture
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

An old open question asks whether the alternating series ∑_{n=1}^∞ (−1)^n n/p_n, with p_n the nth prime, converges. The paper's answer is that it does, conditionally, under the Riemann Hypothesis and a strong prime-k-tuples conjecture, and the route is to replace the series by the damped Riemann–Stieltjes integral I = ∫ $e^{{iπx}}$$e^{{−λx}}$ dψ(x), where ψ is the Chebyshev function. Because the smooth oscillation gives α = 1 and RH makes ψ approximately 1/2-Hölder, Young's criterion applies and the integral exists. Splitting ψ(x) = x + R(x) with R(x) = O($x^{{1/2}}$) yields an explicit main term and an error O($λ^{{−3/2}}$), and the same bound makes the convergence absolute for λ > 3/2. If correct, this gives an analytic explanation of the numerical value near −0.052161 that computations have suggested.

What carries the argument

The central object is the Riemann–Stieltjes integral I = ∫_1^∞ g dψ, with g(x) = $e^{{iπx}}$$e^{{−λx}}$ a damped oscillation and ψ the Chebyshev function. Its load-bearing property is Young's criterion: if f is α-Hölder and G is β-Hölder with α + β > 1, then ∫ f dG exists. With α = 1 and β = 1/2 the criterion is satisfied; the line of reasoning is then completed by the decomposition ψ = x + R with R = O($x^{{1/2}}$), which turns the error term into a Laplace-type integral with value O($λ^{{−3/2}}$).

What would settle it

Take actual prime data, compute the damped integral I(λ) = ∫ $e^{{iπx}}$$e^{{−λx}}$dψ(x) numerically for λ = 1.4 (just below 3/2) and λ = 1.6 (just above), and compare the growth of the error term with the claimed O($λ^{{−3/2}}$) and with the claimed divergence below 3/2. Also check the Hölder premise directly: near a prime power p^k, |ψ(p^k+) − ψ(p^k−)| ≈ log p, which violates |ψ(x) − ψ(y)| ≤ C|x − y|^{1/2} as the two points approach the jump, so a smoothed surrogate would be needed.

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Extended reading notes

Core claim

The paper's central claim is that the convergence of the alternating Erdős series can be decided by an integral: writing ψ for the Chebyshev function and g(x) = $e^{{iπx}}$$e^{{−λx}}$, the series is represented as I = ∫ g dψ. Under the Riemann Hypothesis, ψ(x) = x + O($x^{{1/2}}$), so ψ is taken to be 1/2-Hölder continuous, while g is 1-Hölder continuous; Young's criterion α + β > 1 then guarantees that I exists. Integration by parts and the error bound R(x) = O($x^{{1/2}}$) give I = $e^{{−λ}}$(λ − iπ)^{−2} + O($λ^{{−3/2}}$), and the paper concludes that I is absolutely convergent for λ > 3/2. The author presents this as confirmation that the original series converges conditionally, conditional on RH and the strong prime-k-tuples conjecture that previous work used.

Load-bearing premise

Everything depends on the assumption that the alternating prime series can be written as an integral against the Chebyshev function and that, under the Riemann Hypothesis, that function obeys a square-root Hölder smoothness bound; if either is false, the Young-criterion step does not apply.

Editorial extensions

If this is right

  • If the integral representation is valid, the alternating Erdős series converges conditionally under the Riemann Hypothesis and a strong prime-k-tuples conjecture.
  • The damped integral admits the explicit evaluation I = e^{−λ}(λ − iπ)^{−2} + O(λ^{−3/2}), giving a concrete numerical target near −0.052161.
  • For damping λ > 3/2 the convergence is absolute, so rearrangements of the underlying prime-indexed terms do not alter the limit.
  • The λ > 3/2 threshold is forced by the O(x^{1/2}) error in the Chebyshev function; any improvement in that error would automatically lower the damping threshold.

Reading between the lines

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

  • The paper does not supply a proof that the limit λ → 0 of the damped integral equals the undamped series; connecting them would need an Abel/Tauberian-type argument that is not written out.
  • The assumption that ψ is 1/2-Hölder is difficult to reconcile with ψ having jumps at prime powers; a natural repair is to smooth the von Mangoldt sum before applying Young's criterion.
  • A direct check is feasible: compute partial sums of ∑ (−1)^n n/p_n up to N = 10^8 and compare running averages with −0.052161; agreement would support, disagreement would refute, the claimed conditional convergence.
  • The same machinery suggests that other alternating sums over primes with a smooth weight, such as ∑ (−1)^n (p_{n+1} − p_n)^θ / n^{1/2+ε}, can be studied by choosing a counting function whose error term matches the desired Hölder exponent.
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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 absolute convergence of the Riemann-Stieltjes integral I = ∫_1^∞ e^{iπx}e^{-λx} dψ(x) for λ > 3/2 under the Riemann Hypothesis, and on this basis concludes that it has "confirmed, using an alternative approach, the potential convergence" of the alternating Erdős series S = Σ_{n=1}^∞ (-1)^n n/p_n. The method introduces a damped oscillatory test function g(x) = e^{iπx}e^{-λx}, applies a Hölder/Young criterion, and splits the integral into a main term and an error term. The central steps are the asserted representation S = ∫ g dψ, the claim that the Chebyshev function ψ is 1/2-Hölder continuous under RH, and the derivation of the threshold λ > 3/2. I find that the asserted representation is false, the Hölder-continuity claim is false, and the damping parameter is not removable, so the main conclusion about the original series is unsupported.

Significance. The alternating Erdős series is a genuine open problem, and Tao's conditional result is relevant background. If the proposed integral method were correct, it would provide a new technique for this and similar prime-indexed series. However, the paper's central identity does not connect the Stieltjes integral to the series S, the regularity assumption on ψ is violated by the actual step function, and the convergence of the damped integral does not survive the limit λ→0. The manuscript therefore does not establish any new result about the convergence of S. Credit is due for identifying the relevant literature and for attempting a Stieltjes-integral formulation, but the technical core is invalid.

major comments (5)
  1. [Section 1, 'Thus, we represent the series as...'] The representation S = ∫_1^∞ g(x) dψ(x) is false. Since ψ(x) = Σ_{p^k ≤ x} log p, the Stieltjes integral equals Σ_{p,k} e^{iπ p^k} e^{-λ p^k} log p, whose n-th prime contribution is e^{iπ p_n} e^{-λ p_n} log p_n, not (-1)^n n/p_n. For n = 2 and p_2 = 3, the series term is +2/3 while the integral contributes -e^{-3λ} log 3; no value of λ makes these equal. The sign in the integral is determined by the residue class of the prime p_n, not by the index n, and the weight is log p_n, not n/p_n. Consequently, all subsequent analysis of I concerns a different series of prime powers and cannot confirm convergence of S.
  2. [Section 1, 'Abel summation formula'] The quoted Abel summation formula, Σ a_n = ∫ f dA - lim_{x→∞} f(x)A(x), is misstated: the standard formula requires an evaluation term at the lower endpoint plus the appropriate sign convention, and A(x) must be the partial-sum function of the a_n. The manuscript replaces A by the Chebyshev function without proving that the partial sums of (-1)^n n/p_n are related to ψ. This is not a local typo; it is the step that allegedly converts the discrete series into the Stieltjes integral, and it is never justified.
  3. [Section 2.2] The assertion that ψ(x) = x + O(x^{1/2}) under RH implies |ψ(x) - ψ(y)| ≤ C|x - y|^{1/2} is false. The Chebyshev function is a step function with jumps of size log p at prime powers. For x = p and y = p - ε with ε→0, |ψ(p) - ψ(p-ε)| = log p, which cannot be bounded by C ε^{1/2}. Thus ψ is not Hölder continuous of any positive order, and the Young-criterion argument in §2.4 and the fractional Sobolev embedding in §2.9 do not apply.
  4. [Section 2.1, Theorem 2.1] The proof of Theorem 2.1 does not establish existence of the Riemann-Stieltjes integral. It bounds one Riemann sum by the product of p- and q-variation seminorms, but existence requires that Riemann sums converge to a common limit as the partition is refined, which does not follow from boundedness of the variations. Moreover, the spaces W^p and W^q are never defined, and the condition 1/p + 1/q > 1 is not used in the proof. The later Young criterion involving α + β > 1 is a different condition and cannot be invoked without the false Hölder continuity of ψ.
  5. [Sections 2.8 and 2.12] Even if the integral representation were correct, the parameter λ is not a removable regularization: §2.8 gives I ∼ -1/(λ - iπ) as λ→0+, which diverges as the damping vanishes, whereas the original series has no damping. The threshold λ > 3/2 derived in §2.12 is an artifact of bounding |R(x)g'(x)| by C x^{1/2} λ x^{-λ-1} and requiring convergence of the resulting integral; it is not a property of the original Erdős series. The paper therefore does not address the undamped question.
minor comments (5)
  1. [Throughout] There are many typos and inconsistent notations, including 'Erd{\omicron}s', 'H{\omicron}lder', and the abstract's 'g(x)=x(−1)x' versus the later 'g(x)=e^{iπx}e^{-λx}'.
  2. [Section 2.12] The expression g(x) = (-1)^x e^{-λx} is not well-defined for non-integer x, and the derivative g'(x) = (-1)^x(-λ x^{-λ-1}) does not match either this expression or the earlier complex exponential definition.
  3. [Section 2.11] The evaluation I1 = e^{(iπ-λ)}/(λ-iπ)^2 appears incorrect for the Laplace integral; the standard formula for ∫_1^∞ x e^{(iπ-λ)x} dx contains an additional factor involving 1/(λ-iπ), so the numerical value of the main term is not reliable.
  4. [Conclusions] The conclusion states that convergence holds 'assuming a suitably strong version of the Hardy-Littlewood prime tuples conjecture', but the body assumes the Riemann Hypothesis and never invokes Conjecture 1.3; the relationship between the two assumptions is not discussed.
  5. [References] Several reference entries have broken line breaks in their URLs (e.g., [10], [11], [14]), and the numerical value -0.052161 is cited without a source or computational detail.

Circularity Check

2 steps flagged · score 8.0 of 10

The convergence result is built into the inserted damping factor, and the asserted integral representation of the Erdős series is not derived.

  1. self definitional [Section 1, integral representation (p. 3)]
    "To facilitate the analysis, we approximate the alternating sequence (−1)^n with a smooth, oscillatory function. A suitable candidate is: g(x) = e^{iπx}e^{−λx}, where λ>0 is a damping factor that ensures the function decays for large x ... Thus, we represent the series as: S = ∫_1^∞ g(x)dψ(x)."

    The 'representation' is asserted, not derived from S = Σ(−1)^n n/p_n. Direct evaluation of the Stieltjes integral gives Σ_p e^{iπp}e^{−λp} log p (plus prime-power terms for ψ), a damped sum whose signs attach to the prime p, not to the index n, and whose weights are log p, not n/p_n. The exponential damping e^{−λx} is not present in S. Proving convergence of this integral therefore concerns a different, artificially damped prime-weighted series; the conclusion about S is contained in the redefinition of S as the integral.

  2. other [Sections 2.4–2.8 and Conclusions]
    "we prove that I converges absolutely for λ > 3/2, based on sharp bounds for the error terms. ... I ∼ −1/(λ−iπ), as λ→0+. ... we have confirmed, using an alternative approach, the potential convergence of the alternating Erdős series"

    The absolute convergence of I is guaranteed by the inserted factor e^{−λx} in g: |g'| decays exponentially and ψ(x)=O(x), so the integral converges for every λ>0. As λ→0+ the integral diverges like 1/λ, showing that the claimed convergence is an artifact of the damping chosen in g and not a property inherited from the original undamped Erdős series. The λ>3/2 threshold is itself the product of an incorrect derivative computation, and the true Laplace tail ∫x^{1/2}e^{−λx}dx needs no threshold beyond λ>0. Announcing this as confirmation of the original series announces a property put into the construction by definition.

full rationale

The paper's central inference is circular: it defines a damped integral I = ∫ e^{iπx}e^{−λx} dψ(x), proves that this damped integral converges (trivially, because of the exponential decay), and then presents that convergence as confirming the convergence of the undamped Erdős series S = Σ(−1)^n n/p_n. The bridge between I and S is the asserted identity S = ∫ g dψ, which is never derived and is false: the Stieltjes integral accumulates terms e^{iπp}e^{−λp} log p over primes, whereas S has terms (−1)^n n/p_n indexed by n. Even the Hölder regularity of ψ claimed in §2.2 is unsupported — ψ is a step function with jumps at primes, so |ψ(x)−ψ(y)|≤C|x−y|^{1/2} does not follow from RH — but the decisive circularity is that the convergence result is an input to the construction of g, not an output about S. The conclusion therefore reduces, by the paper's own definitions, to the damped integral's built-in convergence.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central claims rest on the Riemann Hypothesis, an unproved conjecture, and on an asserted but never derived Riemann-Stieltjes representation of the series. The regularity assumption on psi is false for a step function. No free parameters are fitted to data; the only adjustable parameter is the damping factor lambda, which is chosen ad hoc.

free parameters (1)
  • lambda
    A damping parameter introduced in g(x) = e^{i pi x} e^{-lambda x}; the original series has no such factor, so the parameter is ad hoc and the resulting integral convergence does not transfer to the series.
assumptions (3)
  • ad hoc to paper The Riemann Hypothesis holds, so psi(x) = x + O(x^{1/2}).
    The paper assumes this unproved conjecture to claim Hölder regularity for psi; it is stated in Section 2.2 as a given.
  • ad hoc to paper The original series S can be represented as the Riemann-Stieltjes integral integral_1^infinity g(x) d psi(x).
    No valid derivation is provided; the quoted Abel summation formula in Section 1 is misstated and does not establish this equivalence.
  • domain assumption The Chebyshev function psi is Hölder continuous of order 1/2.
    This is asserted in Section 2.2 but is false because psi is a step function with jumps at prime powers, and discontinuous functions cannot be Hölder continuous.

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Cite this review

Pith. "Pith review of Holder continuity of an alternating Erdos series on prime K-tuples." pith.science (2026). https://pith.science/paper/K3QOKTDZ

@misc{pith2026250506242,
  author       = {Pith},
  title        = {Pith review of: Holder continuity of an alternating Erdos series on prime K-tuples},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K3QOKTDZ}},
  note         = {Machine review of arXiv:2505.06242}
}
read the original abstract

This open problem, first posed by Erd{\omicron}s, was further explored by Terence Tao. Tao work shows that the series can converge conditionally, but only under a sufficiently strong form of the Hardy-Littlewood conjecture for k-primary pairs. Based on this, we offer a new method leading to a representation of the series as a Riemann-Stieltjes integral or a tightly coupled prime counting function. We rigorously analyze this integral by decomposing it into principal and error terms, applying integration by parts in the Stieltjes sense, and defining the error terms. Assuming the Riemann hypothesis, we investigate the H{\omicron}lder continuation of {\psi}(x) in the asymptotic form {\psi}(x) = x+O(x 1/2), and introduce a test function g(x) = e^( i{\pi}x) e^( -{\lambda}x) , which is smooth and Lipschitz. Applying Young's criterion, we show that the integral converges. Moreover , we prove that the integral converges perfectly for {\lambda} > 3 2 , based on sharp bounds on the error terms. Our results are supported by fractional Sobolev integrations and justify the use of Young's inequality under generalized Holder conditions.

Discussion (0). Continue with ORCID to comment.

Reference graph

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