{"id":"a9645789-faa3-4368-810b-3eda8a516624","arxiv_id":"2505.06242","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper claims a conditional proof of convergence for the alternating Erdős series, but the proof relies on an invalid integral representation and a false Hölder continuity assertion.","lead":"This paper tries to prove convergence of an alternating series over prime numbers by converting it into a Riemann-Stieltjes integral and using Hölder continuity conditions. The central claims are unsupported, because the integral representation is invalid and the Chebyshev function is not Hölder continuous.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed representation S = ∫ e^{iπx}e^{-λx} dψ(x) is never derived and is false: the Stieltjes integral sums log p with sign (-1)^p, not (-1)^n n/p_n, so proving convergence of the damped integral says nothing about the Erdős series.","rationale":"The reader's REJECT verdict is correct. The central bridge from the Erdős series to the damped Riemann-Stieltjes integral is the single most load-bearing step, and it fails: the Stieltjes integral with respect to the Chebyshev function is a sum over prime powers with weights log p and sign (-1)^p, not the target series. This is an internal inconsistency, not a disagreement with consensus; no amount of RH or Hardy-Littlewood input repairs it. The analysis of I may establish convergence of an auxiliary damped prime-power series, but that object diverges as the damping λ tends to 0 and is unrelated to S. Therefore the paper's central claim is unsupported, and no adjustment to the reader's verdict is needed.","tokens_in":6937,"tokens_out":9182,"duration_ms":92766,"concrete_test":"Fix n=2. The Erdős term is 2/3. In the claimed integral the jump at p_2=3 is g(3)Δψ(3)=e^{3πi}e^{-3λ}log3 = -e^{-3λ}log3, which cannot equal 2/3 for any λ>0. If the authors intend an Abel transform rather than pointwise equality, the test is to write the standard Abel partial-summation identity for S_N explicitly, define the associated measure, and exhibit the exact or estimated steps leading to ∫_1^∞ g dψ. Without such a derivation, the central identity is both unsupported and contradicted by direct computation of the Stieltjes sum.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the asserted identity S = ∫_1^∞ g(x)dψ(x), g(x)=e^{iπx}e^{-λx}, introduced in Section 1 ('Thus, we represent the series as...') without derivation. The identity is false. Since ψ(x)=Σ_{p^k≤x} log p, direct evaluation gives ∫_1^∞ g dψ = Σ_{p^k} e^{iπ p^k}e^{-λ p^k} log p. The n-th prime contributes e^{iπ p_n}e^{-λ p_n} log p_n, whereas the n-th term of S is (-1)^n n/p_n. Pointwise, for n=2 and p_2=3, S contributes +2/3 while the integral contributes e^{3πi}e^{-3λ}log3 = -e^{-3λ}log3; no λ makes them equal. The sign is tied to the prime value p_n rather than the index n, and the weight is log p_n rather than n/p_n. The damping is not removable: §2.8 gives I ~ -1/(λ-iπ) as λ→0+, so the analyzed object diverges as damping is removed, whereas the original series has no damping. Thus convergence of I for λ>3/2 concerns a different series of prime powers and cannot confirm convergence of S. The additional claim that the step function ψ is 1/2-Hölder is also unsupported, but the representation failure is already decisive.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":7355,"tokens_out":7489,"duration_ms":73228,"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":[{"comment":"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.","section":"Section 1, 'Thus, we represent the series as...'"},{"comment":"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.","section":"Section 1, 'Abel summation formula'"},{"comment":"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.","section":"Section 2.2"},{"comment":"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.","section":"Section 2.1, Theorem 2.1"},{"comment":"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.","section":"Sections 2.8 and 2.12"}],"minor_comments":[{"comment":"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}'.","section":"Throughout"},{"comment":"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.","section":"Section 2.12"},{"comment":"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.","section":"Section 2.11"},{"comment":"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.","section":"Conclusions"},{"comment":"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.","section":"References"}],"recommendation":"reject","confidential_remarks":"This manuscript is not suitable for publication in its current form. The central identity S = ∫ g dψ is false and the Hölder-continuity claim for ψ is also false; these are load-bearing errors in the core approach rather than presentation issues. Even a substantial revision would need to replace the claimed representation with a correct relation between the Erdős series and a Stieltjes integral, which is not achieved here. I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe short take: this should be desk-rejected. The central identity is wrong, not just insufficiently justified. The proof analyzes a damped prime-power sum that has no direct relationship to the Erdős series.\n\nWhat is worth saying in its favor: the authors do cite Tao's paper [13] and correctly name the Hardy–Littlewood prime tuples conjecture as the assumption under which Tao's conditional result holds. The idea of trying Young's Hölder criterion on a Stieltjes transform of a prime-counting function is at least a fresh approach, though it is applied to the wrong object here.\n\nThe soft spots are load-bearing and elementary. The claimed representation S = ∫_1^∞ g(x)dψ(x) is asserted in Section 1 without derivation. Direct evaluation gives ∫ g dψ = Σ_{p^k} e^{iπ p^k} e^{-λ p^k} log p, a damped sum over prime powers with sign tied to the prime value, not to the index n and weight n/p_n. The quoted Abel summation formula is misstated, and it cannot produce the identity. Also, ψ is a step function with jumps at prime powers, so the 1/2-Hölder condition claimed in §2.2 is false; the asymptotic ψ(x)=x+O(x^{1/2}) under RH does not imply Hölder continuity.\n\nThe paper's own asymptotics confirm that the damping matters. In §2.8, I ∼ −1/(λ−iπ) as λ→0+, so the analyzed integral diverges as the damping is removed; it cannot be a regularized version of S. There is also an internal inconsistency: §2.6 gives I1 = −e^{-λ}/(λ−iπ), while §2.12 gives a different value e^{iπ−λ}/(λ−iπ)^2 and uses a nonsensical formula for g'(x) involving x^{-λ-1}. Theorem 2.1 is not a valid theorem; the Hölder inequality does not produce the claimed bounded variation result.\n\nIn short, the paper does not prove the convergence of the Erdős series. The errors are not subtle; the main representation is false. I would not send this to a referee. It might be suitable for feedback to the authors about the elementary nature of the error, but not for a journal.","headline":"Desk-reject; the central identity is false and the paper actually analyzes a damped prime-power sum, not the Erdős series.","tokens_in":7768,"tokens_out":3602,"would_cite":false,"duration_ms":32733,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N05","40A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["alternating Erdős series","conditional convergence","Riemann-Stieltjes integral","Chebyshev function","Hölder continuity","Riemann hypothesis","prime k-tuples conjecture","Young criterion"],"falsifier":"Take actual prime data, compute the damped integral I(λ) = ∫ $e^{{iπx}}$$e^{{−λx}}$dψ(x) numerically for λ = 1.4 (just below 3/2) and λ = 1.6 (just above), and compare the growth of the error term with the claimed O($λ^{{−3/2}}$) and with the claimed divergence below 3/2. Also check the Hölder premise directly: near a prime power p^k, |ψ(p^k+) − ψ(p^k−)| ≈ log p, which violates |ψ(x) − ψ(y)| ≤ C|x − y|^{1/2} as the two points approach the jump, so a smoothed surrogate would be needed.","tokens_in":6766,"feed_emoji":"🔢","tokens_out":10186,"duration_ms":92061,"temperature":0.7,"pith_summary":"An old open question asks whether the alternating series ∑_{n=1}^∞ (−1)^n n/p_n, with p_n the nth prime, converges. The paper's answer is that it does, conditionally, under the Riemann Hypothesis and a strong prime-k-tuples conjecture, and the route is to replace the series by the damped Riemann–Stieltjes integral I = ∫ $e^{{iπx}}$$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.","feed_headline":"Under Riemann Hypothesis, alternating prime sum converges","feed_subtitle":"A damped integral against the Chebyshev function converges absolutely for λ>3/2 and targets −0.052161.","key_machinery":"The central object is the Riemann–Stieltjes integral I = ∫_1^∞ g dψ, with g(x) = $e^{{iπx}}$$e^{{−λx}}$ a damped oscillation and ψ the Chebyshev function. Its load-bearing property is Young's criterion: if f is α-Hölder and G is β-Hölder with α + β > 1, then ∫ f dG exists. With α = 1 and β = 1/2 the criterion is satisfied; the line of reasoning is then completed by the decomposition ψ = x + R with R = O($x^{{1/2}}$), which turns the error term into a Laplace-type integral with value O($λ^{{−3/2}}$).","core_discovery":"The paper's central claim is that the convergence of the alternating Erdős series can be decided by an integral: writing ψ for the Chebyshev function and g(x) = $e^{{iπx}}$$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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Prior work proving conditional convergence under a quantitative prime-k-tuples conjecture; supplies the conjecture formulation and the benchmark result.","marker":"[13]"},{"why":"Young's inequality for Hölder continuous functions; the criterion α + β > 1 that carries the integral-existence step.","marker":"[14]"},{"why":"Source for the Riemann-Hypothesis asymptotic ψ(x) = x + O(x^{1/2}) used to assign β = 1/2 to the Chebyshev function.","marker":"[1]"},{"why":"Existence theorem for the Riemann-Stieltjes integral used to justify integration by parts.","marker":"[11]"},{"why":"Abel summation formula that connects the discrete series to an integral against a prime-counting function.","marker":"[10]"},{"why":"Fractional Sobolev embedding used to justify the Hölder regularity of ψ within the function-space framework.","marker":"[12]"}],"fun_headline_variants":["RH gives absolute convergence of Erdős series for λ>3/2","Hölder continuity yields Erdős series convergence under RH","Absolute convergence of Erdős series from RH and λ>3/2","Prime k-tuples and RH settle Erdős series convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["RH gives absolute convergence of Erdős series for λ>3/2","Hölder continuity yields Erdős series convergence under RH","Absolute convergence of Erdős series from RH and λ>3/2","Prime k-tuples and RH settle Erdős series convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000679,"raw_usage":{"total_tokens":3099,"prompt_tokens":969,"completion_tokens":2130,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":2056}},"tokens_in":585,"tokens_out":2130,"duration_ms":15072,"temperature":1.0,"reasoning_tokens":2056,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:58:24.939286+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"The convergence of an alternating series of Erd\\H{o}s, assuming the Hardy--Littlewood prime tuples conjecture","cited_arxiv_id":"2308.07205","evidence_quote":"Prior work proving conditional convergence under a quantitative prime-k-tuples conjecture; supplies the conjecture formulation and the benchmark result."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Young's inequality for Hölder continuous functions; the criterion α + β > 1 that carries the integral-existence step."},{"cited_title":"Chebyshev's bias and generalized Riemann hypothesis","cited_arxiv_id":"1112.2398","evidence_quote":"Source for the Riemann-Hypothesis asymptotic ψ(x) = x + O(x^{1/2}) used to assign β = 1/2 to the Chebyshev function."},{"cited_title":"H\\\"older continuity of functions in the fractional Sobolev spaces: 1-dimensional case","cited_arxiv_id":"2308.06048","evidence_quote":"Fractional Sobolev embedding used to justify the Hölder regularity of ψ within the function-space framework."}],"review_version":1}