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Euler's $\ell$-totients and Riemann hypothesis

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2607.26114 v1 pith:VKFSXR2M submitted 2026-07-28 math.GM

Ahmed Gaber This is my paper · ORCID
classification math.GM MSC 11N3711M2611N64
keywords Euler'sℓ-totientRiemannhypothesisDirichletseriessummatoryfunctionzetazerosMertensasymptoticformulasmeromorphiccontinuation
topics 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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 5 minor

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}).

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 (3)
  1. [Theorem 4.3(1), proof in §4] 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.
  2. [Theorem 4.3(2), proof of analyticity of G(s)] 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.
  3. [Abstract and Introduction, claimed equivalence] 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.
minor comments (5)
  1. [Throughout] Notation is inconsistent: φ_ℓ, ϕ_ℓ, and φℓ are used interchangeably; Φ_ℓ and S_ℓ both denote the summatory function. Please standardize.
  2. [Abstract] Typos: “developes” should be “develops”; “papers” should be “paper”.
  3. [Introduction] “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.
  4. [Theorem 4.3(2), proof] “for k≥1” should be “for ℓ≥1” in the sentence about s = 0 not lying in the half-plane.
  5. [References] Reference [18] contains a typo in the title: “Exponentialsum,men” should be “Exponentialsummen.”

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ℓ-totient criteria are derived from direct Euler-product computations and standard external theorems, not from fitting or self-citation.

full rationale

The paper's central claim, Theorem 4.3(2), is a genuine implication: if the summatory error is O(x^{1−1/(2ℓ)+ε}), then the Dirichlet series F_ℓ(s), computed directly from the definition of φ_ℓ in Theorem 3.1, has no poles with Re(s)>1−1/(2ℓ) except at s=2. Since Corollary 3.3 independently locates poles at s=1+(ρ−1)/ℓ for every nontrivial zeta zero ρ, any zero with β>1/2 would produce a pole in the forbidden half-plane. The error assumption is not used to set parameters or to define F_ℓ; it enters only through analytic continuation of the Mellin transform G(s). No fitted constants are renamed as predictions, and no self-citations are load-bearing; the references to Titchmarsh and Tenenbaum are standard external facts. Theorem 4.3(1) similarly invokes the Littlewood equivalence (RH ⇒ M(x)=O(x^{1/2+η})) as an external theorem and uses partial summation. The proof of part (1) does contain a real mathematical error—the replacement of ∑_{m≤X}m by X²/2+X/2+O(1) with X=⌊x/d^ℓ⌋ drops a fractional-part term that is not absorbed by the displayed error—but that is a correctness issue, not a circularity issue. The apparent cross-reference to 'Corollary 2.5' and the typo 'k≥1' are also editing errors, not circular reasoning. Overall, the claimed derivation chain is self-contained against external benchmarks and does not reduce its conclusions to its hypotheses.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No free parameters are fitted to data; ℓ is an index and ε is an arbitrary small constant. The proof leans on standard zeta facts, the Littlewood equivalence, and Mellin-transform theory. The only new object, φ_ℓ, is explicitly constructed and not used as a hidden assumption.

assumptions (4)
  • standard math Standard analytic properties of ζ(s): Euler product for Re(s)>1, meromorphic continuation, trivial zeros at negative even integers, none in Re(s)≥1 except pole at 1
    Used throughout Theorem 3.1, Corollary 3.3 and Remark 4.4; taken from classical texts.
  • standard math Littlewood equivalence: RH is equivalent to M(x)=O(x^{1/2+η}) for every η>0
    Invoked in proof of Theorem 4.3(1) to estimate T(y); cited to Titchmarsh [17].
  • standard math Functional equation of ζ and zero symmetry (ρ zero ⇒ 1−ρ zero)
    Used in final step of Theorem 4.3(2) to upgrade β≤1/2 to β=1/2.
  • standard math Mellin/Perron framework: summatory error O(x^{α+ε}) implies associated Dirichlet series has analytic continuation to Re(s)>α (after subtracting main pole)
    This is the structural principle behind G(s) analytic in Theorem 4.3(2); the paper proves the specific instance via uniform convergence.
invented entities (1)
  • Euler ℓ-totient function φ_ℓ independent evidence
    purpose: Central arithmetic function whose summatory function provides the RH criteria
    Not an ad hoc hidden entity: it is explicitly defined by φ_ℓ(n)=n∏_{p^ℓ|n}(1−1/p) and its elementary properties (multiplicativity, combinatorial interpretation, Euler product) are derived in the paper.

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Pith. "Pith review of Euler's $\ell$-totients and Riemann hypothesis." pith.science (2026). https://pith.science/paper/VKFSXR2M

@misc{pith2026260726114,
  author       = {Pith},
  title        = {Pith review of: Euler's $\ell$-totients and Riemann hypothesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VKFSXR2M}},
  note         = {Machine review of arXiv:2607.26114}
}
abstract

This paper develops a new analytic framework for investigating the Riemann hypothesis. For each fixed integer $\ell \ge 1$, define Euler's $\ell$-totient function $\varphi_\ell$ by \[ \varphi_\ell(n):=n\prod_{\substack{p\ \mathrm{prime}\\ v_p(n)\ge \ell}}\left(1-\frac{1}{p}\right), \] and its summatory function by \[ \Phi_\ell(x):=\sum_{n\le x}\varphi_\ell(n). \] An analytic study of the generalized Euler $\ell$-totient function $\varphi_\ell(n)$ is carried out, including its Euler product representation, meromorphic continuation, and pole structure. For each $\ell$, necessary and sufficient criteria for the Riemann hypothesis are established in terms of the asymptotic behavior of $\Phi_\ell(x)$.

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Reference graph

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