{"id":"c063ca58-8187-412c-8661-84448a971df6","arxiv_id":"2607.26114","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For ℓ≥1, if the summatory ℓ-totient function Φℓ(x) deviates from its main term by O(x^{1−1/(2ℓ)+ε}) for every ε>0, then the Riemann hypothesis is true.","lead":"Ahmed Gaber introduces a family of generalized Euler totient functions indexed by an integer ℓ and derives a new sufficient condition: if their summatory function has a certain error bound, the Riemann hypothesis follows. The paper also claims a converse (RH implies a linear error bound), but that direction is not proven correctly as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.3(1) is not proven: the derivation drops the fractional-part term -x∑ μ(d){x/d^ℓ}/d, so the claimed necessary O(x) bound fails to follow; the sufficient direction (2) appears sound.","rationale":"The reader's weakest assumption identifies the exact flaw I independently found: the proof of Theorem 4.3(1) invalidly replaces the floor value X by x/d^ℓ in the main terms while dropping the error −xθ_d/d, which accumulates to O(x log x). My own expansion of the exact inner sum confirms this. The sufficient direction (Theorem 4.3(2)) is a standard Mellin-transform argument and is correct; the possible lack of simplicity of zeta zeros does not undermine it because the existence of a pole at s=1+(ρ−1)/ℓ is enough, even if the pole order is higher. Thus the central claim of a necessary-and-sufficient criterion is not fully established, but the paper's sufficiency result remains a legitimate modest contribution. This is exactly the reader's conditional position, so no verdict change is needed.","tokens_in":10580,"tokens_out":22551,"duration_ms":168190,"concrete_test":"Re-derive the expression for Φ_ℓ(x) in Theorem 4.3(1) without omitting the fractional-part terms. Specifically, start from the exact identity Φ_ℓ(x) = ∑_{d≤x^{1/ℓ}} μ(d)/d · d^ℓ N(N+1)/2, substitute N = x/d^ℓ − {x/d^ℓ}, and verify whether the term −x∑_{d≤x^{1/ℓ}} μ(d){x/d^ℓ}/d appears. For ℓ=1, numerically evaluate S(x) = ∑_{d≤x} μ(d){x/d}/d for x = 10^k, k = 3,...,9; if |S(x)| grows like log x rather than remaining bounded, the missing term is not O(x) and the proof of Theorem 4.3(1) fails as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4.3(1) asserts that\n\n∑_{m≤X} m = X²/2 + X/2 + O(1)   (X = ⌊x/d^ℓ⌋)\n\nand, after the 'usual substitution' n = d^ℓ m, obtains\n\nΦ_ℓ(x) = x²/2 ∑_{d≤x^{1/ℓ}} μ(d)/d^{ℓ+1} + x/2 ∑_{d≤x^{1/ℓ}} μ(d)/d + O(∑_{d≤x^{1/ℓ}} d^{ℓ-1}).\n\nThis is incorrect. With N = ⌊x/d^ℓ⌋ and θ_d = {x/d^ℓ}, the exact inner sum is\n\nd^ℓ N(N+1)/2 = x²/(2d^ℓ) + x/2 − x θ_d + d^ℓ(θ_d²/2 − θ_d/2).\n\nMultiplying by μ(d)/d and summing gives an extra term\n\nE(x) = −x ∑_{d≤x^{1/ℓ}} μ(d) θ_d / d,\n\nwhich is not absorbed by O(∑ d^{ℓ-1}) = O(x); in general it is O(x log x). The paper gives no argument that this term is O(x) (or o(x)) under RH. Consequently the claimed necessary direction, 'RH ⇒ Φ_ℓ(x) = x²/(2ζ(ℓ+1)) + O(x)', is not established. The sufficient direction (Theorem 4.3(2)) does not rely on this step and appears valid: it only needs every zero ρ to yield a pole of F_ℓ(s) at s = 1+(ρ−1)/ℓ, which holds regardless of zero simplicity (the simplicity assertion in Corollary 3.3 is unproven but not needed).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces, for each integer ℓ ≥ 1, an ℓ-totient function φ_ℓ(n) = n ∏_{p^ℓ | n} (1 − 1/p), studies its elementary properties and its Dirichlet series F_ℓ(s) = ζ(s−1)/ζ(ℓ(s−1)+1), and gives a meromorphic continuation with poles at s = 2, at s = 1 + (ρ−1)/ℓ for every nontrivial zero ρ of ζ, and at points arising from trivial zeros. The main result, Theorem 4.3, claims two directions: (1) RH implies Φ_ℓ(x) = x²/(2ζ(ℓ+1)) + O_{ℓ,ε}(x); (2) if Φ_ℓ(x) = x²/(2ζ(ℓ+1)) + O_{ℓ,ε}(x^{1−1/(2ℓ)+ε}) for every ε > 0, then RH holds. The proof of direction (2) uses a Mellin-transform identity and pole cancellation; the proof of direction (1) uses Mertens’ bound and a partial-summation estimate for the tail T(x^{1/ℓ}).","tokens_in":11058,"tokens_out":8300,"duration_ms":75650,"significance":"If Theorem 4.3(2) is correct as written, it supplies a new sufficient criterion for RH, parameterized by ℓ, and the Mellin-pole mechanism is transparent and free of fitted constants. The paper’s elementary analysis of φ_ℓ, including the Euler product, Möbius-inversion formula, and the tail bound, is mostly sound. However, the claimed necessary direction (Theorem 4.3(1)) is not proven: the displayed derivation drops a fractional-part term, and the paper gives no argument that this term is O(x) under RH. Thus the advertised biconditional is not established, although the sufficient direction appears sound. The paper would be a valuable contribution if the necessary direction were repaired or if the claims were honestly reduced to the one-way criterion.","major_comments":[{"comment":"The proof contains a load-bearing algebraic error. After setting N = ⌊x/d^ℓ⌋ and θ_d = {x/d^ℓ}, the exact inner sum is d^ℓ N(N+1)/2 = x²/(2d^ℓ) + x/2 − x θ_d + d^ℓ(θ_d²/2 − θ_d/2). Multiplying by μ(d)/d and summing over d ≤ x^{1/ℓ} produces the additional term −x ∑_{d≤x^{1/ℓ}} μ(d) θ_d / d. This term is not of size O(∑ d^{ℓ−1}) = O(x); in general it is O(x log x) by the trivial bound, and the manuscript supplies no cancellation argument under RH. Consequently, the displayed expression after the sentence “Using ∑_{m≤X} m = X²/2 + X/2 + O(1)” omits a term that can dominate the intended O(x) error. The subsequent bounds on the tail T(x^{1/ℓ}) and on ∑ μ(d)/d do not fill this gap, so the claimed implication RH ⇒ Φ_ℓ(x) = x²/(2ζ(ℓ+1)) + O(x) does not follow from the proof as written. This is a central claim of the abstract and introduction, not a peripheral estimate.","section":"Theorem 4.3(1), proof in §4"},{"comment":"This direction appears sound, but it relies on the notation “Corollary 2.5” when referring to the pole structure of F_ℓ(s); the intended reference is Corollary 3.3. More substantively, the proof of the pole-cancellation at s = 2 is correct, and the mapping of zeros ρ with Re(ρ) > 1/2 to poles with Re(1+(ρ−1)/ℓ) > 1−1/(2ℓ) is valid. However, the paper should state explicitly that the simplicity of those poles is not needed for the argument — only the existence of a pole at each such s — because the simplicity claim in Corollary 3.3(2) is asserted without full justification for the nontrivial-zero case, though the cancellation argument given is essentially sufficient.","section":"Theorem 4.3(2), proof of analyticity of G(s)"},{"comment":"The abstract and introduction describe “necessary and sufficient criteria for the Riemann hypothesis.” Since the necessary direction (Theorem 4.3(1)) is not established by the given proof, the manuscript’s central claim is stronger than what is proved. If the gap in part (1) cannot be repaired, the theorem should be restated as a one-way sufficient criterion and the abstract, introduction, and title-level claims should be adjusted accordingly. The sufficient criterion itself is a meaningful result and should not be buried by the overclaim.","section":"Abstract and Introduction, claimed equivalence"}],"minor_comments":[{"comment":"Notation is inconsistent: φ_ℓ, ϕ_ℓ, and φℓ are used interchangeably; Φ_ℓ and S_ℓ both denote the summatory function. Please standardize.","section":"Throughout"},{"comment":"Typos: “developes” should be “develops”; “papers” should be “paper”.","section":"Abstract"},{"comment":"“see see [17, p. 370]” has a duplicated “see”. Also the phrase “Euler’s ℓ-totient functionφ ℓ” and “generalized Euler totient functionϕ ℓ(n)” should be unified.","section":"Introduction"},{"comment":"“for k≥1” should be “for ℓ≥1” in the sentence about s = 0 not lying in the half-plane.","section":"Theorem 4.3(2), proof"},{"comment":"Reference [18] contains a typo in the title: “Exponentialsum,men” should be “Exponentialsummen.”","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The sufficient direction (Theorem 4.3(2)) is a credible and interesting contribution, and the Mellin-transform argument is clean. The necessary direction, however, has a genuine gap that is central to the advertised equivalence. I would not reject the paper outright, because the one-way criterion may be salvaged by removing or repairing part (1). If the author cannot supply a correct proof of the O(x) bound under RH, the manuscript should be revised to present only the sufficient criterion and to temper the abstract/introduction accordingly. I also note the paper is posted in math.GM; the result, if properly scoped, could be acceptable in a number-theory journal after revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper is worth refereeing, but not as it stands. The ℓ-totient functions and their Dirichlet series are a legitimate generalization of the classical totient, and the sufficient criterion in Theorem 4.3(2) — an O(x^{1−1/(2ℓ)+ε}) error in the summatory function forces RH — is a valid Mellin-transform argument. The necessary direction, RH ⇒ O(x), is plausible but the proof as written does not establish it; the stress-test note is correct.\n\nWhat is actually new: the definition φ_ℓ(n)=n∏_{p^ℓ|n}(1−1/p) is a natural thinning of Euler’s totient, and the Euler product ζ(s−1)/ζ(ℓ(s−1)+1) is correct. The pole map s_ρ=1+(ρ−1)/ℓ is a clean observation: the zeros’ real parts are rescaled to 1+(β−1)/ℓ, so under RH they land on 1−1/(2ℓ). The argument for part (2) uses standard Mellin inversion cleanly, shows the error integral is analytic in the half-plane, and derives the contradiction from the pole locations. For ℓ>1 this is a new sufficient criterion.\n\nThe soft spots are localized. Theorem 4.3(1) replaces the exact inner sum with X²/2+X/2+O(1) for X=⌊x/d^ℓ⌋ and drops the fractional part. The exact expression contains a term −xθ_d, which after summation gives −x∑_{d≤x^{1/ℓ}} μ(d){x/d^ℓ}/d. No argument shows this is O(x) under RH; the trivial bound is O(x log x). So the claimed linear error does not follow from the displayed derivation. This is a genuine gap, not a cosmetic one. Also, Corollary 3.3 asserts the poles from non-trivial zeros are simple; that requires the unproven simplicity of zeta zeros. The sufficient part does not need simplicity, only existence of the poles, so the gap is not load-bearing there.\n\nThe paper is not a lost cause. The sufficient direction alone is a modest, reproducible result (the Mellin computation and the pole map check out). And part (1) can likely be repaired with a standard Perron/Mellin argument: under RH the poles lie at Re=1−1/(2ℓ), so the error should be O(x^{α+ε}), which is stronger than O(x). The authors just chose the wrong route. A referee should ask for that correction or for the claim to be downgraded.\n\nBottom line: send to peer review. It deserves a serious referee; the fix is likely routine, and the ℓ-totient family is worth having in the literature. I would bring it to a reading group if you want a concrete example of a mostly-correct RH criterion with a very teachable flaw.","headline":"The sufficient criterion in Theorem 4.3(2) is a real, clean result; the necessary direction in Theorem 4.3(1) is not proven as written because the derivation drops the fractional-part error, so the iff framing outruns the evidence.","tokens_in":11522,"tokens_out":19282,"would_cite":true,"duration_ms":149516,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N37","11M26","11N64"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the size of the error in Euler's ℓ-totient summatory function is a two-way test for the Riemann hypothesis: RH implies a linear error, and a power-saving error for any ℓ forces RH.","keywords":["Euler's ℓ-totient","Riemann hypothesis","Dirichlet series","summatory function","zeta zeros","Mertens function","asymptotic formulas","meromorphic continuation"],"falsifier":"Compute the inner sum used in the proof of Theorem 4.3(1): for X = 100.5, ∑_{m≤X} m = 5050 while X²/2+X/2 = 5100.375, a gap of about 50 that grows linearly with X. This directly falsifies the O(1) replacement of the floor by the real number in the derivation of the necessary direction; for the sufficient direction, the decisive test would be to exhibit or construct a zeta zero with Re(ρ)>1/2 and check that it creates a pole of F_ℓ(s)/s in Re(s)>1−1/(2ℓ), contradicting the claimed analytic continuation.","tokens_in":10476,"feed_emoji":"🧮","tokens_out":9801,"duration_ms":79887,"temperature":0.7,"pith_summary":"This paper introduces Euler's ℓ-totient, φ_ℓ(n) = n ∏_{p^ℓ|n} (1−1/p), which counts integers up to n that avoid every prime whose ℓ-th power divides n, and studies its summatory function Φ_ℓ(x). The main theorem is a two-way bridge between the size of the error term in Φ_ℓ(x) and the location of the zeros of the Riemann zeta function. If the Riemann hypothesis is true, the paper argues Φ_ℓ(x) = x²/(2ζ(ℓ+1)) + O(x) for every fixed ℓ. Conversely, for any fixed ℓ, if the error term is O(x^{1−1/(2ℓ)+ε}) for every ε>0, then all nontrivial zeta zeros must lie on the critical line, proving RH. The interest is that this gives a new, ℓ-parameterized family of arithmetic statements any one of which, if established, would settle the Riemann hypothesis.","feed_headline":"Small totient-sum error forces all zeta zeros onto the critical line","feed_subtitle":"A power-saving error bound for any ℓ-totient would prove the Riemann hypothesis.","key_machinery":"Euler's ℓ-totient φ_ℓ(n) = n∏_{p^ℓ|n}(1−1/p) with summatory function Φ_ℓ(x). The load-bearing identity is the Dirichlet series F_ℓ(s)=ζ(s−1)/ζ(ℓ(s−1)+1): the denominator ζ(ℓ(s−1)+1) transforms each zeta zero ρ into a simple pole of F_ℓ at s=1+(ρ−1)/ℓ, making the zero set visible as a pole pattern. The sufficient direction then runs through the integral representation G(s)=∫_1^∞(Φ_ℓ(x)−x²/(2ζ(ℓ+1)))x^{−s−1}dx = F_ℓ(s)/s − (1/(2ζ(ℓ+1)))/(s−2), whose analyticity forces a pole-free half-plane. This machinery converts a growth estimate for Φ_ℓ into an analyticity statement for F_ℓ, and hence into a restriction on the real parts of zeta zeros.","core_discovery":"The central object is the Dirichlet series F_ℓ(s) = ∑ φ_ℓ(n)n^{-s} = ζ(s−1)/ζ(ℓ(s−1)+1). Its nontrivial poles occur at s = 1+(ρ−1)/ℓ for each nontrivial zero ρ of ζ, so the location of zeta zeros is encoded in the poles of F_ℓ. The paper's key analytic step is to write Φ_ℓ(x) = x²/(2ζ(ℓ+1)) + E_ℓ(x) and form the Mellin-type integral G(s)=∫_1^∞ E_ℓ(x)x^{-s−1}dx, which equals F_ℓ(s)/s − (1/(2ζ(ℓ+1)))/(s−2). If E_ℓ(x)=O(x^{1−1/(2ℓ)+ε}) for all ε>0, then G(s) is analytic in the half-plane Re(s)>1−1/(2ℓ), so F_ℓ(s) can have no poles there. But every zero with Re(ρ)>1/2 would produce exactly such a pole; hence no such zero can exist, and the functional equation forces all zeros onto the critical l","pith_inferences":["Editorial: The derivation of Theorem 4.3(1) has a gap: replacing ⌊x/d^ℓ⌋ by x/d^ℓ inside ∑_{m≤X} m = X(X+1)/2 introduces an error O(X) per term, not O(1); summed over d this contributes O(x log x), so the claimed 'RH ⇒ O(x)' bound does not follow from the displayed computation. The sufficient direction (Theorem 4.3(2)) does not rely on this step.","Editorial: If the floor terms were handled by a sharper estimate, the linear-error consequence would still be consistent with existing heuristics and classical results for φ_1; the principal novelty—the power-saving sufficient criterion—stands independently.","Editorial: The pole map s = 1+(ρ−1)/ℓ suggests a general technique: for any Dirichlet series whose numerator and denominator are zeta-type factors, the same rescaling yields a family of RH-type sufficient criteria in terms of summatory functions.","Editorial: The ℓ-family offers a concrete numerical programme: compute Φ_ℓ(x) for several ℓ and large x to test whether the error exponent can be brought below 1−1/(2ℓ); failure for any ℓ would indicate the existence of an off-critical zero rather than a flaw in the analytic continuation."],"forward_implications":["If the paper's sufficient criterion is correct, proving the error bound O_{ℓ,ε}(x^{1−1/(2ℓ)+ε}) for any single ℓ≥1 is enough to prove the Riemann hypothesis.","Because the criterion holds for every ℓ, the family provides many independent arithmetic formulations; at least one would show a detectable failure if RH were false.","Under RH, the paper's necessary part would imply the summatory ℓ-totient error is O(x) for each fixed ℓ, which is much sharper than the classical O(x log x) bound for Euler's totient.","The pole map s=1+(ρ−1)/ℓ gives a concrete numerical signature: off-critical zeros would appear as poles of F_ℓ in the half-plane Re(s)>1−1/(2ℓ), a feature that could be searched for computationally.","The φ_ℓ functions interpolate between Euler's totient and Möbius-like structures, potentially connecting to existing criteria expressed in terms of arithmetic functions."],"fun_headline_variants":["Totient error bound would settle Riemann hypothesis","ℓ-totient sum error ties to zeta zeros","One error bound for totient sums proves RH","How a totient sum error condition forces RH","Zeta zeros pinned by totient error term"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof of Theorem 4.3(1) assumes that replacing the exact sum ∑_{m≤X} m = X(X+1)/2 with X²/2 + X/2 + O(1) is valid when X = ⌊x/d^ℓ⌋ is replaced by the real number x/d^ℓ; for non-integer X the remainder is actually O(X), not O(1), so that step is load-bearing and false as written.","fun_headline_variants_meta":{"raw":{"variants":["Totient error bound would settle Riemann hypothesis","ℓ-totient sum error ties to zeta zeros","One error bound for totient sums proves RH","How a totient sum error condition forces RH","Zeta zeros pinned by totient error term"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000274,"raw_usage":{"total_tokens":1495,"prompt_tokens":785,"completion_tokens":710,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":641}},"tokens_in":529,"tokens_out":710,"duration_ms":7260,"temperature":1.0,"reasoning_tokens":641,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T01:18:26.341308+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the inner sum used in the proof of Theorem 4.3(1): for X = 100.5, ∑_{m≤X} m = 5050 while X²/2+X/2 = 5100.375, a gap of about 50 that grows linearly with X. This directly falsifies the O(1) replacement of the floor by the real number in the derivation of the necessary direction; for the sufficient direction, the decisive test would be to exhibit or construct a zeta zero with Re(ρ)>1/2 and check that it creates a pole of F_ℓ(s)/s in Re(s)>1−1/(2ℓ), contradicting the claimed analytic continuation.","supporting_citations":[],"review_version":1}