{"id":"77aab3e6-3b8c-4ef8-9bd6-31a004befa2f","arxiv_id":"2411.19259","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Riemann hypothesis is equivalent to the boundedness of the normalized divisor sum sigma_{1/2}(n)/(n^{1/2} exp(li((log n)^{1/2}))).","lead":"This paper proves new equivalent conditions for the Riemann hypothesis in terms of how large the sum of 1/2-th powers of divisors of n can be. If true, it adds a fresh, purely arithmetic formulation of RH to the toolbox of analytic number theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified. The apparent vulnerability in Lemma 7.4(2)—possible cancellation of the logarithmic singularities in ~F+—survives inspection: the coefficient κ m_ρ/(ρ(ρ−κ)) is nonzero, so the Landau argument for (ii)⇒(i) stands.","rationale":"The paper proves a sharp equivalence: no zeros of ζ in Re(s)>κ if and only if aκ(n) is bounded, if and only if limsup aκ(n) equals −Bκζ(κ). The proof structure is coherent, and the strongest claim in Corollary 2 follows as a special case. I focused on the converse direction, since that is where a hidden cancellation or missing logarithmic term would break the argument. In Lemma 7.4(2), the two logarithmic contributions combine to κm_ρ/(ρ(ρ−κ))≠0; the branch point at s=1 cancels; and the use of Landau's theorem is legitimate because under the contradiction hypothesis F+ is nonnegative and O(u^{Θ−1−δ}), so its transform has abscissa Θ−δ and is holomorphic in Re(s)>Θ−δ. The additional auxiliary estimates check out: Lemma 4.5 depends on the cancellation between Ei and X^{ρ−κ}/(ρ log X), which converts the zero sum into a convergent ∑|ρ|^{-2} expression; Lemma 5.1's Mertens constants are correct; and the convexity argument in §6 supplies the matching upper bound over all n. No unsupported identity or unjustified interchange surfaced. The reader's ACCEPT verdict with moderate confidence is appropriate; the only reason not to have full confidence is that the Omega-lemma is intricate and not machine-checked, but that is not a demonstrated defect.","tokens_in":23536,"tokens_out":47734,"duration_ms":405496,"concrete_test":"Independently re-derive Lemma 7.4(2): compute −∫_s^∞ (1/z ζ′/ζ(z)+2^{1−z}/(z−1))dz near ρ by contour deformation and verify that the coefficient of log(1/(σ−β)) is −m_ρ/ρ, so that the combined coefficient from the two singular terms is κm_ρ/(ρ(ρ−κ)). If this coefficient came out zero, the Landau contradiction would fail and the converse direction would collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on the converse (ii)⇒(i), whose pivot is Lemma 7.4(2). I checked this genuinely load-bearing step rather than assuming it: near a zero ρ=β+iγ with β>κ, the transform ~F+ has a logarithmic branch point. The term from −log((s−1)ζ(s)) contributes +m_ρ/(ρ−κ)·log(1/(σ−β)); the integral term contributes −m_ρ/ρ·log(1/(σ−β)). The total coefficient is κm_ρ/(ρ(ρ−κ)), which is nonzero for κ∈[1/2,1), so the singularity is non-removable. The branch point at s=1 arising from the transform of li(u^{1−κ}) is cancelled by the pole of ζ(s) inside log((s−1)ζ(s)), so Lemma 7.4(1) needs no extra excluded point. I also checked the supporting estimates: Lemma 4.5 relies on the cancellation between Ei((ρ−κ)log X) and X^{ρ−κ}/(ρ log X), which removes the otherwise divergent ∑|ρ|^{-1}; the Mertens constants in Lemma 5.1 are consistent once the log(log y_2/log x) parentheses are restored; and the convexity upper bound in §6 gives the matching limsup. This is an honest non-finding: the weakest point is the Omega-lemma, but on inspection it holds.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a two-way equivalence between zero-free regions for the Riemann zeta-function and the maximal order of generalized divisor functions. For κ in [1/2,1), Theorem 1 states that the absence of zeros in Re(s)>κ is equivalent to the boundedness of a_κ(n) and also to Ramanujan's limiting formula for limsup a_κ(n). Corollary 2 specializes this to a necessary and sufficient condition for the Riemann hypothesis in terms of a_{1/2}(n). The proof develops κ-superior highly composite numbers, connects σ_κ on these numbers to the partial Euler product at s=κ, and then treats the Euler product by explicit formulas. The converse direction uses a Landau-type Omega-estimate; the delicate singularity analysis is in Lemma 7.4. Theorem 3 gives an analogous criterion in terms of the partial Euler product E_1(X). Overall the paper is carefully written and the central equivalence is proved in both directions.","tokens_in":23909,"tokens_out":8422,"duration_ms":72396,"significance":"If correct, the paper supplies a new explicit arithmetic criterion for the Riemann hypothesis: boundedness of the concrete function a_{1/2}(n) is equivalent to all nontrivial zeros lying on Re(s)=1/2. The proof uses standard analytic number theory tools, carries explicit error terms, and does not introduce fitted constants or special assumptions beyond the definition of κ-SHCNs. The most delicate point is the Omega-estimate in Proposition 7.1, whose justification rests on the singularity analysis in Lemma 7.4. I independently checked the coefficient κ m_ρ/(ρ(ρ−κ)) in (7.9); it is nonzero for κ∈[1/2,1), so the logarithmic branch point is not cancelled and the Landau argument stands. The paper thus makes a solid contribution to the literature on Ramanujan's maximal order problem and its connections to the Riemann hypothesis.","major_comments":[],"minor_comments":[{"comment":"The phrase 'the sum of the 1/2-th powers of divisors function' is slightly awkward; consider referring to 'the divisor function σ_{1/2}(n)' throughout.","section":"Abstract and Section 1"},{"comment":"The values of the limiting expressions in condition (iii) are stated only in equation (1.3); repeating them explicitly in the theorem statements would make the paper more self-contained.","section":"Theorem 1 and Corollary 2"},{"comment":"Lemma 8.1 is quoted from [1] without proof; since it is used in the proof of Theorem 3, a short parenthetical remark that the lemma is unconditional and independent of RH would improve readability.","section":"Section 8, Lemma 8.1"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is mathematically sound. The only point that deserves a second independent check before publication is the singularity analysis in Lemma 7.4, which I have verified carefully. The use of [1] for Lemma 8.1 is not load-bearing for Theorem 1. I am happy to accept after the minor editorial revisions are made."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new content is the converse direction (ii)=>(i) in Theorem 1: boundedness of a_kappa(n) forces Re(s)<=kappa for nontrivial zeros of zeta. That direction is genuinely new; the forward implication (i)=>(iii) was already in Ramanujan's suppressed work and in the author's own previous paper. The E1 criterion in Theorem 3 is a clean by-product. I looked hard at the singularity analysis in Lemma 7.4, because if the logarithmic branch contributions canceled there, the Omega-estimate in Proposition 7.1 would collapse. They do not cancel: the combined coefficient is kappa*m_rho/(rho(rho-kappa)), which is nonzero for kappa in [1/2,1). So the Landau argument goes through. I also checked the supporting points that seemed most fragile: the cancellation in Lemma 4.5 between Ei((rho-kappa)log X) and X^(rho-kappa)/(rho log X), the Mertens constants in Lemma 5.1, and the convexity argument in Section 6. Everything is consistent. There is no circular step and no fitted constant. The self-citations are legitimate because the cited lemmas are either re-derived or standard. The soft spots are not fatal. The paper is another equivalent formulation of RH of the Robin/Lagarias type; it does not get closer to RH itself, and the impact is confined to analytic number theory. The kappa-SHCN machinery is heavy, and the Omega-lemma is subtle enough that I would want a referee to check it line by line. Section 2, especially the six exponentials theorem part, is tangential; the paper could be shorter without it, but the material is not wrong. Who is this for? Analytic number theorists who collect RH equivalences and people working on maximal orders of arithmetic functions. The main theorem is new, the proof is detailed, and the delicate step holds up on inspection. It deserves a serious referee, not a desk rejection. I would accept it after a careful check of the estimates in the Omega-lemmas.","headline":"A genuine new converse direction in an RH equivalence, with the load-bearing Omega-lemma surviving inspection; worth refereeing carefully.","tokens_in":24379,"tokens_out":2435,"would_cite":true,"duration_ms":22070,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N56","11M26"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Riemann hypothesis holds exactly when one divisor function is bounded.","keywords":["divisor functions","maximal order","Riemann zeta-function","Riemann hypothesis","zero-free region","superior highly composite numbers","partial Euler product","explicit formula"],"falsifier":"Compute the identity (7.4) numerically for several real $s>\\kappa$: evaluate both sides using tabulated values of $\\zeta(s)$ and its zeros; a mismatch would show that Lemma 7.3 or the singularity analysis of Lemma 7.4 is incorrect, which would remove the lower-bound argument proving the implication (ii)$\\Rightarrow$(i).","tokens_in":23369,"feed_emoji":"","tokens_out":19623,"duration_ms":145275,"temperature":0.7,"pith_summary":"This paper proves that the maximal order of the divisor-sum function $\\sigma_\\kappa(n)$ is governed exactly by the location of the zeros of the Riemann zeta-function. For each $\\kappa \\in [1/2,1)$, the paper shows that $\\zeta(s)$ has no zeros with $\\mathrm{Re}(s)>\\kappa$ if and only if the normalized function $a_\\kappa(n)$ is bounded, and if and only if its limsup equals the explicit constant $-B_\\kappa\\zeta(\\kappa)$. Taking $\\kappa=1/2$ gives a new necessary and sufficient condition for the Riemann hypothesis: $a_{1/2}(n)$ is bounded exactly when every nontrivial zero lies on the critical line. A parallel equivalence is proved for a partial Euler product. The upshot is that a deep analytic question about zeros is shown to be encoded in the ordinary growth of an explicit arithmetic function of the divisors of $n$.","feed_headline":"One bounded divisor function is equivalent to the Riemann hypothesis","feed_subtitle":"The paper shows boundedness of the half-power divisor sum holds iff RH does.","key_machinery":"The argument runs through two constructions. First, $\\kappa$-superior highly composite numbers, defined as integers $N$ maximizing $\\sigma_\\kappa(n)/n^{\\kappa(1+\\varepsilon)}$, provide a sparse sequence on which $\\sigma_\\kappa(N)/N^\\kappa$ is asymptotically the partial Euler product $\\prod_{p\\le X}(1-p^{-\\kappa})^{-1}$ with $X$ tied to $\\log N$ (Propositions 3.2 and 5.1). Second, the partial Euler product is analysed by an explicit formula (Proposition 4.4) expressing it as $\\mathrm{li}(X^{1-\\kappa})$ plus a sum over the nontrivial zeros of $\\zeta$ weighted by exponential integrals. Under a zero-free region the zero sum is small and yields the limsup formula; if a zero exists with $\\mathrm{Re}(\\rho)>\\kappa$, a Landau-type $\\Omega_+$ estimate (a lower bound holding infinitely often at the displayed rate, Proposition 7.1) shows the product is unbounded, using a Mellin transform $\\widetilde F_+(s)$ whose logarithmic singularity at such $\\rho$ (Lemma 7.4) forces the lower bound. A convexity inequality (Lemmas 6.2 and 7.5) then transfers the limsup statement from the special sequence to all $n$.","core_discovery":"The central claim is an equivalence: for $\\kappa \\in [1/2,1)$, the statements (i) $\\zeta(s)$ has no zeros in $\\mathrm{Re}(s)>\\kappa$, (ii) $a_\\kappa(n)$ is bounded on $n\\ge 3$, and (iii) $\\limsup_{n\\to\\infty} a_\\kappa(n) = -B_\\kappa\\zeta(\\kappa)$ (with $B_\\kappa=1/\\sqrt{2}$ for $\\kappa=1/2$ and $B_\\kappa=1$ otherwise) are mutually equivalent. In particular, the Riemann hypothesis is equivalent to boundedness of $a_{1/2}(n) = \\sigma_{1/2}(n)/(n^{1/2}\\exp[\\mathrm{li}((\\log n)^{1/2})])$ and also to the identity $\\limsup_{n\\to\\infty} a_{1/2}(n) = -\\zeta(1/2)/\\sqrt{2}$. The paper proves the same two-way statement for a partial Euler product: the Riemann hypothesis holds if and only if $E_1(X)=\\prod_{p\\le X}(1-p^{-1/2})^{-1}/\\exp[\\mathrm{li}(\\vartheta(X)^{1/2})]$ is bounded, and if and only if it converges to $-\\sqrt{2}\\,\\zeta(1/2)$. This transforms a previously known conditional evaluation of the maximal order of $\\sigma_\\kappa(n)$ into a rigorous criterion of equal strength in both directions.","pith_inferences":["The mechanism suggests an analogue for other Dirichlet series with a similar logarithmic singularity at zeros: a suitably normalized divisor sum being bounded could be equivalent to the absence of zeros in a given half-plane.","Because the theorem achieves its limsup along the $\\kappa$-SHCN sequence, a numerical test of the criterion can be restricted to that sparse, explicitly listed set rather than all integers; any persistent overshoot of the predicted constant would be an immediate red flag.","The comparison inequality in Lemma 7.5 keeps the $\\Omega_+$ lower bound from being swamped when passing from the special sequence to all $n$; strengthening it could convert the unboundedness into explicit growth rates in $\\Theta$.","The equivalence is insensitive to the precise normalization: any function differing from $a_\\kappa(n)$ by a factor that is $1+o(1)$ along the SHCN sequence would carry the same criterion, so the specific shape $\\exp[\\mathrm{li}((\\log n)^{1-\\kappa})]$ is natural but not unique."],"forward_implications":["The Riemann hypothesis is equivalent to a single explicit statement about ordinary integers: the function $a_{1/2}(n)$ is bounded for $n\\ge 3$.","The classical maximal-order formula for $\\sigma_\\kappa(n)$ is exactly as sharp as the corresponding zero-free region: each region $\\mathrm{Re}(s)>\\kappa$ free of zeros corresponds to the limsup formula for $a_\\kappa(n)$.","A partial Euler product gives a second equivalent: the Riemann hypothesis holds exactly when $E_1(X)$ is bounded, and exactly when it tends to $-\\sqrt{2}\\,\\zeta(1/2)$.","If a zero with $\\mathrm{Re}(\\rho)>\\kappa$ existed, both $a_\\kappa(n)$ and the partial Euler product would be unbounded with an explicit $\\Omega_+$ rate, so the growth of divisor sums is a direct measure of zero location."],"supporting_citations":[{"why":"The annotated version of the suppressed part of the highly composite numbers paper; it introduced generalised superior highly composite numbers and proved conditionally the limsup formula that this paper turns into an equivalence.","marker":"[23]"},{"why":"The standard text supplying Landau's theorem, the explicit formula for $\\zeta'/\\zeta$, Perron's formula, and the Mertens-type estimates used throughout.","marker":"[13]"},{"why":"The earlier treatment of the Euler product in the critical strip; used for Lemma 8.1 and for rewriting the partial Euler product into explicit formulas.","marker":"[1]"},{"why":"The classical paper establishing the maximal-order results for $\\sigma_\\kappa(n)$ in $\\kappa\\ge 1$ that this paper extends to $\\kappa\\in[1/2,1)$.","marker":"[8]"},{"why":"The original highly composite numbers paper; source of the superior highly composite construction that is generalised in Section 2.","marker":"[20]"},{"why":"The paper supplying the bound $y_k(x)<(kx)^{1/k}$ used in Lemma 3.1 for the size of the sequence of $\\kappa$-superior highly composite numbers.","marker":"[6]"},{"why":"The reference for exponential integral and logarithmic integral identities used in the explicit formula (Proposition 4.4) and its estimates.","marker":"[11]"}],"fun_headline_variants":["Divisor sum bounded iff Riemann hypothesis holds","RH equivalence: bounded half-power divisor sum","Bounded divisor function proves Riemann hypothesis","Riemann hypothesis tied to divisor sum bound","Partial Euler product bounded iff RH true"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The converse direction rests on the claim that near every zero $\\rho$ of $\\zeta$ with $\\mathrm{Re}(\\rho)>\\kappa$, the integral representation of the partial Euler product has a genuine logarithmic blow-up that is not cancelled by any other term; if that blow-up were removable, the lower-bound estimate and hence the whole equivalence would fail.","fun_headline_variants_meta":{"raw":{"variants":["Divisor sum bounded iff Riemann hypothesis holds","RH equivalence: bounded half-power divisor sum","Bounded divisor function proves Riemann hypothesis","Riemann hypothesis tied to divisor sum bound","Partial Euler product bounded iff RH true"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000133,"raw_usage":{"total_tokens":1139,"prompt_tokens":953,"completion_tokens":186,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":122}},"tokens_in":569,"tokens_out":186,"duration_ms":2764,"temperature":1.0,"reasoning_tokens":122,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T10:22:06.511485+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the identity (7.4) numerically for several real $s>\\kappa$: evaluate both sides using tabulated values of $\\zeta(s)$ and its zeros; a mismatch would show that Lemma 7.3 or the singularity analysis of Lemma 7.4 is incorrect, which would remove the lower-bound argument proving the implication (ii)$\\Rightarrow$(i).","supporting_citations":[{"cited_title":"Ramanujan, Highly composite numbers","cited_arxiv_id":null,"evidence_quote":"The annotated version of the suppressed part of the highly composite numbers paper; it introduced generalised superior highly composite numbers and proved conditionally the limsup formula that this paper turns into an equivalence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The standard text supplying Landau's theorem, the explicit formula for $\\zeta'/\\zeta$, Perron's formula, and the Mertens-type estimates used throughout."},{"cited_title":"Akatsuka, The Euler product for the Riemann zeta-func tion in the critical strip, Kodai Math","cited_arxiv_id":null,"evidence_quote":"The earlier treatment of the Euler product in the critical strip; used for Lemma 8.1 and for rewriting the partial Euler product into explicit formulas."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The classical paper establishing the maximal-order results for $\\sigma_\\kappa(n)$ in $\\kappa\\ge 1$ that this paper extends to $\\kappa\\in[1/2,1)$."},{"cited_title":"Ramanujan, Highly composite numbers, Proc","cited_arxiv_id":null,"evidence_quote":"The original highly composite numbers paper; source of the superior highly composite construction that is generalised in Section 2."},{"cited_title":"Caveney, J.-L","cited_arxiv_id":null,"evidence_quote":"The paper supplying the bound $y_k(x)<(kx)^{1/k}$ used in Lemma 3.1 for the size of the sequence of $\\kappa$-superior highly composite numbers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The reference for exponential integral and logarithmic integral identities used in the explicit formula (Proposition 4.4) and its estimates."}],"review_version":1}