{"id":"eff01392-19fd-4aed-bf10-f0da470e1d18","arxiv_id":"2502.02598","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the summatory function of 2^omega(n), the paper proves an unconditional decomposition with error O(x^{8/29+epsilon}) and a conditional decomposition involving zeros of zeta(2s).","lead":"This paper studies the total of 'two to the number of distinct prime factors' over all integers up to x. It gives a sharper error formula unconditionally and an even finer one under a strong Riemann hypothesis.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 is internally inconsistent: with T=x^{10} the vertical integral is O(x^{9/2+ε}), not O(x^ε), so the stated conditional asymptotic does not follow from the proof in Section 3.","rationale":"The reader's verdict is CONDITIONAL, and I agree with that final disposition, but not with the stated weakest assumption. The decisive flaw is not the unproved Lemma 2.4; it is the arithmetic inconsistency in Theorem 1. With T=x^{10}, the vertical contour integral is bounded by x^{9/2+ε}, so the displayed O(x^ε) error cannot be obtained. The natural repair, choosing T=x, preserves an O(x^ε) formula but changes the zero-sum cutoff from x^{10} to x and must be stated explicitly. I do not raise a separate fundamental objection to the strongest claim, Theorem 2: given Lemma 2.4 and Lemma 2.5, the balancing T=x^{21/29} and the resulting O(x^{8/29+ε}) error appear internally coherent, although a small technical point about the exact good height T^* from Lemma 2.4 should also be checked when finalizing the statement. Because a main theorem is unproved as stated but appears repairable, CONDITIONAL revision is the appropriate verdict.","tokens_in":10826,"tokens_out":34334,"duration_ms":360140,"concrete_test":"Recompute the three error terms in (3.1) with T=x^{10}: vertical = O(x^{-1/2}T^{1/2}) = O(x^{9/2+ε}); horizontal terms are O(x^{1/4}T^{-1/2+3ε}) = O(x^{-19/4+ε}) and O(x^{1+ε}T^{-1+3ε}) = O(x^{-9+4ε}); Perron error = O(x^{1+ε}/T) = O(x^{-9+ε}). Since the vertical term alone exceeds O(x^ε), the claimed error cannot follow from the displayed estimates; no numerical experiment is needed.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 3 shifts the contour to Re(s)=-1/2 and bounds the vertical integral by |I1| ≪ x^{-1/2+ε} T^{1/2} after using the functional equation and (1.4). The proof then sets T=x^{10} and concludes an error O(x^{10ε}), i.e., O(x^ε). Substituting T=x^{10} gives |I1| ≪ x^{-1/2}·x^5 = x^{9/2+ε}, which is not O(x^ε). This vertical term also swamps the displayed horizontal terms, which are O(x^{-19/4+ε}) and O(x^{-9+4ε}), and the Perron error O(x^{-9+ε}). Thus Theorem 1's claimed error term is not established by the estimates given. The proof can likely be repaired by choosing T=x, which makes |I1|=O(1) and lowers the zero-sum cutoff from x^{10} to x, but as stated the theorem is unsupported. This is an internal numerical inconsistency in the proof, independent of the cited external Lemma 2.4.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the summatory function of the strongly multiplicative function 2^{\\omega(n)}. The authors write its Dirichlet series as F(s)=\\zeta(s)^2/\\zeta(2s) and apply Perron's formula with a contour shift to \\Re(s)=-1/2. Under a strong Riemann hypothesis they claim, in Theorem 1, an asymptotic formula with leading terms A_1 x\\log x + A_2 x, a sum over residues at zeros of \\zeta(2s) with \\Re(s)=1/4 and |\\Im(2s)|1/2, but for \\sigma=1/4 the argument of \\zeta(2s) has real part exactly 1/2, so the lemma does not apply uniformly on the closed segment. The authors should either justify this boundary behavior explicitly or choose the contour height via a good-height lemma such as Lemma 2.4, as is done in the proof of Theorem 2.","section":"Section 3, estimate of I_2+I_3"},{"comment":"The statement of Theorem 2 fixes the residue-sum cutoff at |\\gamma|\\le x^{21/29}, but the proof uses a height T=T^*/2 where T^* is the good height supplied by Lemma 2.4. Lemma 2.4 only guarantees that T^* lies in an interval [T_0,T_0+T_0^{1/3}], not that T^* equals 2x^{21/29} exactly. Since the number and size of the residues with ordinates between the chosen good height and x^{21/29} are not controlled, replacing the actual contour height by x^{21/29} is not automatic. The theorem should either state the cutoff as the chosen good height or give an additional argument showing that the difference is absorbed in O(x^{8/29+\\varepsilon}).","section":"Section 4, paragraph 'we make a special choice T such that 2T=T^*'"}],"minor_comments":[{"comment":"The intermediate formula contains the term \\zeta(0) from the residue at s=0; this is absorbed into O(x^\\varepsilon) in Theorem 1, but the absorption is not stated explicitly and may confuse readers.","section":"Section 3, Eq. (3.1)"},{"comment":"Since Theorem 2 depends on Lemma 2.4 for the reciprocal of \\zeta on a horizontal line, the authors should give the exact statement and proof location in [8], or include a proof sketch, rather than simply citing the result.","section":"Section 2, Lemma 2.4"},{"comment":"The notation \\gamma_{1/4} in the zero-sum of Theorem 1 is not defined precisely; it should be stated that \\gamma_{1/4}=\\Im(2\\rho) for a zero \\rho of \\zeta(2s).","section":"Theorem 1 and Theorem 2"},{"comment":"There are several typographical issues in the author affiliation and abstract (for example, \"Hyderaba d\" and \"T elangana\"), which should be corrected in production.","section":"Front matter"},{"comment":"Remark 5 lists several further asymptotics that might follow by the same method, but no proofs or precise statements are provided; it should be labeled as speculative or removed.","section":"Remark 5"}],"recommendation":"major_revision","confidential_remarks":"The central unconditional result, Theorem 2, appears methodologically sound modulo the cutoff issue in its statement, while Theorem 1 has a clear numerical inconsistency in the choice T=x^{10}. Both problems are local and repairable, so I do not recommend rejection. The authors should also tighten the use of Lemma 2.3 in the horizontal estimate of Theorem 1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper has one solid result, Theorem 2, and one conditional theorem whose stated error term is not supported by the proof as written. Theorem 1 chooses T=x^10 and then claims an O(x^ε) error, but the vertical integral at Re(s)=-1/2 is O(x^{-1/2}T^{1/2}) = O(x^{9/2+ε}) with that choice. The fix is to take T=x, which makes the vertical term O(1) and the remaining error O(x^ε), but it also changes the zero cutoff from x^10 to x. That is a substantive change to the main term, not a typo, so the theorem statement needs revising.\n\nWhat is genuinely new: Theorem 2 gives, unconditionally, an asymptotic formula with an explicit sum over poles of ζ(s)^2/ζ(2s) and error O(x^{8/29+ε}). That improves the previous O(x^{1/2} exp{-C (log x)^{3/5}/(log log x)^{1/5}}) error term from the literature. The method is standard Perron contour plus existing zeta estimates, but the decomposition is explicit and the exponent 8/29 is competitive (smaller than 131/416 in the divisor problem). No fitted parameters; the residues are derived. The paper is clearly written and the unconditional argument appears coherent.\n\nSoft spots, in proportion: aside from the Theorem 1 error, the proof of Theorem 2 says \"we make a special choice T such that 2T = T*\" of Lemma 2.4. That is confusing. Lemma 2.4 gives a good height T* near T; you would normally pick the contour height to be exactly T*, not half of it. If the authors mean the chosen contour height is T*, then the notation is just awkward. If they mean half, the horizontal bounds at height T are not justified because the 1/ζ bound only holds at T*. Needs clarification. The unconditional proof depends on Lemma 2.4 (an external result from [8]) to control 1/ζ(2s) on horizontal lines; this is cited without proof, which is acceptable, but it is load-bearing. The constant ζ(0) is omitted from the main terms but absorbed into the error; minor.\n\nWho this is for: analytic number theorists working on multiplicative functions and zeta zeros. Theorem 2 deserves publication. Theorem 1, once corrected, is a nice conditional explicit formula. I would send this to a serious referee with a request to fix the T choice in Theorem 1 and clarify the height selection in Theorem 2.","headline":"Theorem 2 is a solid unconditional improvement, but Theorem 1's proof has a clear T=x^10 error that needs fixing before the paper is publishable.","tokens_in":11582,"tokens_out":7682,"would_cite":true,"duration_ms":71335,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes explicit asymptotic formulas for counting integers by distinct prime factors, with an unconditional error term of order $x^{8/29+\\varepsilon}$ and an $O(x^\\varepsilon)$ error under the strong Riemann hypothesis.","keywords":["strongly multiplicative function","2^{\\omega(n)}","Riemann zeta function","Dirichlet series","Perron formula","Riemann hypothesis","explicit formula","zeta zeros"],"falsifier":"Search a range of large $T$ for a height $T^*\\in[T,T+T^{1/3}]$ where both $\\zeta(\\sigma+iT^*)$ and its reciprocal are bounded by $T^\\varepsilon$ for $1/2\\le\\sigma\\le2$; if such a height never exists for some large $T$, the horizontal-line estimate behind the unconditional error term collapses.","tokens_in":10584,"feed_emoji":"🔢","tokens_out":13253,"duration_ms":123314,"temperature":0.7,"pith_summary":"The paper studies the summatory function of $2^{\\omega(n)}$, where $\\omega(n)$ counts the distinct prime factors of $n$; on squarefree integers this function agrees with the divisor function $d(n)$. Its central object is the Dirichlet series $F(s)=\\zeta(s)^2/\\zeta(2s)$, and the paper proves that the partial sums of $2^{\\omega(n)}$ are recovered by collecting the residues of $F(s)x^s/s$ at its poles. The unconditional theorem expresses the sum as $A_1x\\log x+A_2x$ plus an explicit sum over the zeros of $\\zeta(2s)$ in a strip, with an error term $O(x^{8/29+\\varepsilon})$, improving the previous error term of shape $x^{1/2}\\exp\\{-C(\\log x)^{3/5}(\\log\\log x)^{-1/5}\\}$. Under the strong Riemann hypothesis, the zero sum reduces to zeros on the line $\\Re(s)=1/4$ and the error is $O(x^\\varepsilon)$. A sympathetic reader would care because the proof turns a classical counting problem into an explicit-formula problem whose error is governed by the zeros of the zeta function.","feed_headline":"Counting integers by distinct prime factors: error drops to x^{8/29}","feed_subtitle":"The count of integers grouped by distinct prime factors now has error O(x^{8/29+ε}), better than the previous estimate.","key_machinery":"The central object is the Dirichlet series $F(s)=\\sum 2^{\\omega(n)}n^{-s}=\\zeta(s)^2/\\zeta(2s)$. The paper applies Perron's formula to recover the partial sum as a contour integral of $F(s)x^s/s$, then shifts the contour to $\\Re(s)=-1/2$ in a rectangle and collects residues at $s=1$, $s=0$, and the zeros of $\\zeta(2s)$ that remain after cancellations against zeros of $\\zeta(s)^2$. The unconditional improvement comes from choosing the horizontal sides at a special height $T^*\\in[T,T+T^{1/3}]$, where both $\\zeta$ and $1/\\zeta$ are $O_\\varepsilon(T^\\varepsilon)$, and from bounding the integrand with a subconvexity estimate for $\\zeta$; balancing $x^{1/4}T^{-29/84}=x^{1/2}T^{-29/42}$ gives $T=x^{21/29}$ and error $x^{8/29+\\varepsilon}$.","core_discovery":"The paper's central claim is that the distribution of $2^{\\omega(n)}$ is fully encoded, up to a small remainder, by the quotient $F(s)=\\zeta(s)^2/\\zeta(2s)$. Applying Perron's formula and shifting the contour to $\\Re(s)=-1/2$ gives, via Cauchy's residue theorem, the leading terms $A_1x\\log x+A_2x$ plus a sum of residues at the poles of $F(s)x^s/s$ lying in the rectangle; unconditionally the surviving poles are the zeros $\\rho=\\beta+i\\gamma$ of $\\zeta(2s)$ with $0\\le\\beta\\le1/2$ and $0<|\\gamma|\\le x^{21/29}$, and the remainder is $O_\\varepsilon(x^{8/29+\\varepsilon})$. Assuming the strong Riemann hypothesis, all zeros of $\\zeta(2s)$ lie on $\\Re(s)=1/4$ and are simple, so the pole sum runs over that line up to a height controlled by $x$, and the error becomes $O(x^\\varepsilon)$.","pith_inferences":["One consequence the paper leaves implicit is that the final exponent $8/29$ is a function of the input subconvexity exponent: a stronger bound than the $13/42$ estimate for $\\zeta$ would mechanically lower the error term through the same balancing step.","A natural extension would treat $2^{k\\omega(n)}$ for fixed $k\\ge2$, whose Dirichlet series is $\\zeta(s)^k/\\zeta(ks)$; the analogous zero sum would be governed by zeros of $\\zeta(ks)$ with a height cutoff determined by the same balancing identities.","A numerical probe of the explicit zero sum could test whether the true error is closer to $O(x^{1/4+\\varepsilon})$ than to $O(x^{8/29+\\varepsilon})$: averaging the residue sum over dyadic $x$ and checking whether its mean square grows like $x^{1/2+\\varepsilon}$ would support the plausible conjecture in the paper's Remark 2."],"forward_implications":["The new unconditional error exponent $8/29$ is smaller than the exponent $131/416$ appearing in the best-known Dirichlet divisor problem, so this sum is now known more accurately than the analogous divisor sum.","The main fluctuation in the sum is displayed explicitly as a sum of residues at zeros of $\\zeta(2s)$ up to height $x^{21/29}$, instead of being hidden inside an opaque error term.","Under the strong Riemann hypothesis the zero sum collapses to the line $\\Re(s)=1/4$ and the error becomes $O(x^\\varepsilon)$, with the remaining question being the size of the oscillatory zero sum, which the authors conjecture is $O(x^{1/4+\\varepsilon})$.","The contour procedure is designed to transfer to other zeta-quotient Dirichlet series, including those for $\\varphi(n)/n$, $d(n^2)$, $d(n)^2$, and the squareful indicator function."],"supporting_citations":[{"why":"Supplies the previous unconditional error term for the same sum, the standard zeta-function facts used in the residue computation, and the zero-counting bound.","marker":"[1]"},{"why":"Quoted as Lemma 2.5, it provides the subconvexity bound for $\\zeta$ on vertical strips that controls the horizontal integrals in the unconditional proof.","marker":"[2]"},{"why":"Quoted as Lemma 2.4, it yields the special height $T^*$ where both $\\zeta$ and $1/\\zeta$ are $O_\\varepsilon(T^\\varepsilon)$; this is the load-bearing input for the unconditional horizontal-line estimates.","marker":"[8]"},{"why":"Supplies the functional equation, the size of $\\chi(s)$, and the Riemann-hypothesis bounds for $\\zeta$ and $1/\\zeta$ used in Theorem 1 and in the vertical-line estimates.","marker":"[10]"}],"fun_headline_variants":["Prime factor counting error improves to O(x^{8/29+ε})","Under SRH, error for 2^{ω(n)} sum is O(x^ε)","Error cut to x^(8/29+ε) for distinct-prime counting","New bound for sum of 2^{ω(n)}: O(x^{8/29+ε})","Integer divisor counts: unconditional error O(x^{8/29+ε})"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The unconditional error term rests on an imported lemma, not proved in the paper, that just above any large height there is a line where both the zeta function and its reciprocal are bounded by a small power of the height; if that lemma is false, the $O(x^{8/29+\\varepsilon})$ bound does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Prime factor counting error improves to O(x^{8/29+ε})","Under SRH, error for 2^{ω(n)} sum is O(x^ε)","Error cut to x^(8/29+ε) for distinct-prime counting","New bound for sum of 2^{ω(n)}: O(x^{8/29+ε})","Integer divisor counts: unconditional error O(x^{8/29+ε})"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001633,"raw_usage":{"total_tokens":6446,"prompt_tokens":849,"completion_tokens":5597,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":5486}},"tokens_in":465,"tokens_out":5597,"duration_ms":42253,"temperature":1.0,"reasoning_tokens":5486,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:45:55.045474+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search a range of large $T$ for a height $T^*\\in[T,T+T^{1/3}]$ where both $\\zeta(\\sigma+iT^*)$ and its reciprocal are bounded by $T^\\varepsilon$ for $1/2\\le\\sigma\\le2$; if such a height never exists for some large $T$, the horizontal-line estimate behind the unconditional error term collapses.","supporting_citations":[{"cited_title":"Ivi´ c,The Riemann Zeta-Function: Theory and Applications , Dover Publi- cations, Inc., Mineola, New Y ork, 2003","cited_arxiv_id":null,"evidence_quote":"Supplies the previous unconditional error term for the same sum, the standard zeta-function facts used in the residue computation, and the zero-counting bound."},{"cited_title":"Bourgain, Decoupling, exponential sums and the Riemann zeta function , J","cited_arxiv_id":null,"evidence_quote":"Quoted as Lemma 2.5, it provides the subconvexity bound for $\\zeta$ on vertical strips that controls the horizontal integrals in the unconditional proof."},{"cited_title":"Ramachandra and A","cited_arxiv_id":null,"evidence_quote":"Quoted as Lemma 2.4, it yields the special height $T^*$ where both $\\zeta$ and $1/\\zeta$ are $O_\\varepsilon(T^\\varepsilon)$; this is the load-bearing input for the unconditional horizontal-line estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the functional equation, the size of $\\chi(s)$, and the Riemann-hypothesis bounds for $\\zeta$ and $1/\\zeta$ used in Theorem 1 and in the vertical-line estimates."}],"review_version":1}