On a relation to the Riemann Hypothesis and an analytic part for the divisor function

Hideto Iwata

REVIEW 3 major objections 3 minor 6 references

On a relation to the Riemann Hypothesis and an analytic part for the divisor function

T0 review · 3 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper claims a Riemann Hypothesis bound for the analytic part of the divisor-sum error term, analogous to the Euler totient case.

desk verdict Theorem 1.2.1 contradicts the paper's own definition of E^AN_{σ1}, and the proof drops a dominant error term; reject. read the letter →

arxiv 2601.11052 v1 pith:F5OQNOH2 submitted 2026-01-16 math.NT

Hideto Iwata This is my paper · ORCID

classification math.NT MSC 11M2611N3711A25

keywords divisorfunctionsummatoryanalyticpartarithmeticRiemannHypothesisMellintransformVolterraintegralequationerrorterm

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

The paper attempts to prove that, under the Riemann Hypothesis, the analytic part E^AN_{σ1}(x) of the error term in the asymptotic formula for the summatory function of the divisor function σ_1(n) satisfies E^AN_{σ1}(x) ≪ x^{δ'} exp(log x / log log x), where δ'=max{1/2,δ} for any 0<δ<1. This would extend a known RH-based bound from the Euler totient function to the divisor function. The author also argues that, unlike the totient case, this bound is one-way: the analytic part for σ_1 cannot be used to characterize the Riemann Hypothesis. The proof uses a Mellin transform identity and a contour shift under RH.

What carries the argument

The central object is the analytic part E^AN_{σ1}(x)=1/2 g_2(x)+x/2(log x+2γ−1), with g_2(x)=∑_{n≥1}{x/n}², arising from the Volterra integral equation decomposition of the divisor-summatory error. The carrying identity is the Mellin transform ∫_1^∞ E^AN_{σ1}(x)x^{-s-1}dx = π²/12·1/(s−2)+ζ(s)ζ(s−1)/(s(1−s))+O(1), which after inverse Mellin transform and contour shift reduces the bound to estimating ζ(s)ζ(s−1) in a strip using RH. The distinguishing feature is the factor ζ(s)ζ(s−1) with no ζ(s) in the denominator, which the author argues prevents the equivalence reversal seen for the totient function.

What would settle it

Compute E^AN_{σ1}(x) = 1/2∑_{n≥1}{x/n}² + x/2(log x+2γ−1). Since the square series is nonnegative, E^AN_{σ1}(x) ≥ x/2(log x+2γ−1). For x large this term is ≫ x log x, which grows faster than x^{δ'} exp(log x / log log x) for any δ'<1. For instance, at x=10^6 the explicit x log x term alone exceeds any such claimed bound by a factor that diverges as x^{1/2}/exp(o(log x)).

Watch

Extended reading notes

Core claim

In the paper's own terms, the central discovery is the bound (1.2.1): under the Riemann Hypothesis, for every 0<δ<1 and x≥e^e, |E^AN_{σ1}(x)| ≪ x^{δ'} exp(log x / log log x) with δ'=max{1/2,δ}. Here E^AN_{σ1}(x) = 1/2∑_{n≥1}{x/n}² + x/2(log x+2γ−1) is the analytic part of the error term E_{σ1}(x)=∑_{n≤x}σ_1(n)−(π²/12)x², obtained through the Volterra integral equation decomposition. The proof derives the bound from the Mellin transform of E^AN_{σ1}, which is ζ(s)ζ(s−1)/(s(1−s)) plus pole terms, by shifting the contour to the critical line and applying RH bounds for ζ(s).

Load-bearing premise

The proof rests on the unstated assumption that the O(x^{3+δ}) remainder from the inverse Mellin transform in Lemma 3.1.2 can be discarded in the final estimate; that remainder, if kept, overwhelms the bound the theorem claims.

Editorial extensions

If this is right

If the bound holds, the analytic part of the divisor-summatory error would be O(x^{δ'} exp(log x / log log x)), placing it below the full error's leading terms.
A direct corollary would be the bound E^AN_{σ1}(x) ≪_ε x^{δ'+ε} for every ε>0.
Unlike the totient case, no converse implication to the Riemann Hypothesis would follow from this analytic part.
For arithmetic functions in the Volterra framework whose Dirichlet series has ζ(s) in the numerator but not the denominator, the analytic part would not yield an RH characterization.

Reading between the lines

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

The definition of E^AN_{σ1}(x) explicitly includes the term x/2(log x+2γ−1), which alone grows like x log x; since g_2(x) is nonnegative, E^AN_{σ1}(x) ≥ x/2(log x+2γ−1) for large x, so the claimed sublinear bound is inconsistent with the paper's own definition regardless of the analytic machinery.
The proof discards an O(x^{3+δ}) error term from the inverse Mellin transform in Lemma 3.1.2; this remainder is larger than the bound being proved, so the argument as written would not establish the theorem even if the x log x term were absent.
The paper's negative observation about the absence of an RH-equivalence for σ_1 is plausible as a structural remark: the Mellin transform involves ζ(s)ζ(s−1) rather than ζ(s−1)/ζ(s), so the pole at s=2 is not cancelled by a denominator ζ(s).

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The headline: Theorem 1.2.1 is false as stated, and the proof doesn't survive contact with its own definition. The analytic part E^AN_{σ1}(x) is defined in (2.1.15) as (1/2)g_2(x) + (x/2)(log x + 2γ − 1). Since g_2 ≥ 0, E^AN ≥ (x log x)/2, which for any δ′<1 is larger than x^{δ′} exp(log x/log log x) once x is large. That alone kills the claimed bound; no RH input is needed. The stress-test note is correct.

The paper does have a legitimate framework: the decomposition of the divisor-summatory error term into an arithmetic part x f_2(x) and an analytic part follows Kaczorowski–Wiertelak’s Volterra approach, and the author correctly observes that the RH equivalence for Euler’s totient does not carry over to σ_1. The Mellin transform in Lemma 3.1.1 is standard, and the writing is generally clear.

But the problems are not minor. The proof of Theorem 1.2.1 in §4.1 ends with an O(x^{3+δ}) term after the contour shift. That term dominates x^{δ′} exp(log x/log log x) for any δ<1, and the proof silently drops it. There’s also an unexplained discrepancy in the residue calculation in Lemma 3.1.2. The final theorem is therefore unsupported even if the definition were altered.

I’d reject this as it stands. It deserves a serious referee only if the author fixes the definition/theorem mismatch; right now the contradiction is too immediate. I wouldn’t cite it, and the reading-group value is mainly as a caution about checking a claimed bound against the object’s explicit formula.

Referee Report

3 major / 3 minor

Summary. The paper studies an 'analytic part' E^AN_{σ1}(x) for the error term in the summatory function of σ_1(n), using a Volterra integral-equation framework previously developed by the author. The main result (Theorem 1.2.1) claims that, under the Riemann Hypothesis, E^AN_{σ1}(x) ≪ x^{δ'} exp(log x / log log x) for x ≥ e^e, with δ' = max{1/2, δ} and 0 < δ < 1 arbitrary. A corollary (Theorem 1.2.2/1.2.3) is derived as E^AN_{σ1}(x) ≪ x^{δ'+ε}. The derivation consists of an explicit formula for E^AN_{σ1}(x), Mellin-transform identities, contour deformation, and estimates of ζ(s)ζ(s−1) under the Riemann Hypothesis.

Significance. If the main theorem were correct, it would provide a conditional bound analogous to the known result for the Euler-totient analytic part, which is a known equivalent of the Riemann Hypothesis. The author explicitly notes (Remark 4.2.1) that no such equivalence is obtained for σ_1(n). The paper's explicit decomposition and Mellin transforms are clearly presented; however, the central claim is immediately falsified by the paper's own definition of E^AN_{σ1}(x), making the result untenable as stated. The proof also contains an unresolved dominating error term in the final displayed estimate.

major comments (3)

[§2.1, Eq. (2.1.15) vs. Theorem 1.2.1] The definition in (2.1.15) gives E^AN_{σ1}(x) = (1/2)g2(x) + (x/2)(log x + 2γ − 1), with g2(x) ≥ 0. Therefore E^AN_{σ1}(x) ≥ (x/2)(log x + 2γ − 1), which is ≫ x log x. Theorem 1.2.1 asserts E^AN_{σ1}(x) ≪ x^{δ'} exp(log x/log log x) with δ' = max{1/2, δ} < 1 for every 0 < δ < 1. For any such δ', the ratio x^{1−δ'} log x · exp(−log x/log log x) tends to infinity as x → ∞, so the claimed upper bound is impossible. This is a direct contradiction from the paper's own equations and does not depend on the later estimates or on the Riemann Hypothesis.
[§4.1, display after (4.1.13)] The proof of Theorem 1.2.1 ends with E^AN_{σ1}(x) ≪ x^δ exp(log x/log log x) + x^{1/2} exp(log x/log log x) + O(x^{3+δ}). The term O(x^{3+δ}) is retained in this equation and then silently omitted when the theorem is stated. Since x^{3+δ} dominates the claimed bound x^{δ'} exp(log x/log log x) for every 0 < δ < 1 and all large x, the proof as written does not establish the theorem. This is a load-bearing gap independent of the contradiction in (2.1.15).
[§4.2 vs. Theorem 1.2.3] The corollary is stated as Theorem 1.2.3 but its proof is headed 'Proof of Theorem 1.2.2'. This is a numbering inconsistency that should be corrected if the paper is revised, though it does not affect the mathematical content.

minor comments (3)

[Abstract/Introduction] The abstract says 'explicit bounds', but the constants in Theorem 1.2.1 are implicit; the dependence on δ is also left implicit. Clarify whether the bound is meant to be uniform in δ or valid for each fixed δ.
[§2.1, Eq. (2.1.14)] The decomposition E_{σ1}(x) = E^AR_{σ1}(x) + E^AN_{σ1}(x) + O(x^{1/2}) in (2.1.14) contains an error term, so the names 'arithmetic part' and 'analytic part' are only approximate. This is worth stating explicitly, especially since the paper draws an analogy with the exact decomposition (1.1.6).
[§4.1, Fact 4.1.1 bound] In the line before (4.1.6), the bound for ζ(s)ζ(s−1) is written with a factor |t|^{1−η} and then replaced by |t| exp(log|t|/log log|t|); the transition is valid but slightly compressed. A sentence explaining the inequality would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's theorem is false against its own definition, but falsehood is not circularity.

full rationale

The paper's central claim is a conditional bound under the Riemann Hypothesis. The proof assumes RH (normal for a conditional theorem) and uses standard zeta-function estimates (Lemma 1.2.2 from [3], Fact 4.1.1 from [6]); these are not derived from the claimed result. E^AN_{σ1} is defined explicitly in (2.1.15) as 1/2 g2(x) + x/2(log x + 2γ − 1), with g2(x) = ∑ {x/n}^2 ≥ 0. Hence a direct lower bound E^AN_{σ1}(x) ≥ x/2(log x + 2γ − 1) follows from the paper's own definition, and this contradicts the claimed upper bound for δ' < 1. Moreover, in the proof at (4.1.4) an explicit O(x^{3+δ}) term is displayed and then silently omitted before 'Taking δ′ = max{1/2,δ}'; since x^{3+δ} dominates all displayed terms for 0<δ<1, the proof is invalid as written. These are fatal correctness errors, not circular reasoning: the conclusion is not being assumed or fitted, and no prediction is equivalent by construction to an input. The author's previous work [1] supplies the Volterra-equation framework and auxiliary lemma, but the main estimate is not obtained by citing [1] for its truth; it is derived independently (and incorrectly). The self-citation is therefore not load-bearing in the circularity sense. Remark 4.2.1 itself concedes that a corresponding RH-equivalence is unavailable for σ1(n), which further indicates the paper does not disguise an input as an output. Overall, the derivation is self-contained enough for circularity analysis, despite being mathematically false.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper's contribution is a conditional bound under RH; the main unproved inputs are the RH assumption and standard zeta bounds from the literature. No new entities are introduced. The parameter δ is a hand-chosen contour parameter that controls the final exponent.

free parameters (1)

δ = arbitrary, 0<δ<1
A contour-shift parameter used to define the paths in Lemma 3.1.2 and the rectangle in §4.1; the final exponent δ'=max{1/2,δ} depends on it, and no specific value is fixed or fitted.

assumptions (6)

domain assumption Riemann Hypothesis
Assumed in Theorem 1.2.1 to apply Lemma 1.2.2 and Fact 4.1.1; the theorem is conditional on it.
domain assumption Bound for ζ(s−1)/ζ(s) under RH (Lemma 1.2.2, from [3])
Used to estimate the shifted contour in §4.1; the constants t0 and A are not derived in this paper.
standard math log ζ(s) bound under RH (Fact 4.1.1, from [6])
Used to bound ζ(s)ζ(s−1) on the line σ=1/2+η.
standard math Mellin inversion theorem (Fact 3.1.3, Titchmarsh)
Justifies the inverse Mellin transform in Lemma 3.1.2.
standard math Dirichlet series ∑σ1(n)n^{-s}=ζ(s)ζ(s−1)
Used in Lemma 3.1.1 and (3.1.5).
domain assumption Volterra equation framework from author's previous work [1]
Theorem 2.1.1 is quoted from [1] and underpins the definition of the arithmetic/analytic parts.

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Cite this review

Pith. "Pith review of On a relation to the Riemann Hypothesis and an analytic part for the divisor function." pith.science (2026). https://pith.science/paper/F5OQNOH2

@misc{pith2026260111052,
  author       = {Pith},
  title        = {Pith review of: On a relation to the Riemann Hypothesis and an analytic part for the divisor function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F5OQNOH2}},
  note         = {Machine review of arXiv:2601.11052}
}

read the original abstract

Let phi(n) denote the Euler totient function. We study the analytic part associated with the summatory function of sigma_1(n) and obtain explicit bounds under the Riemann Hypothesis. In particular, we establish an upper bound of order x^{delta'} exp((log x)/(log log x)), where delta' = max(1/2, delta).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references

[1]

Iwata, On the solution of the V olterra integral equation of the second type for the error term in an asymptotic formula for arithmetic functions, Adv

H. Iwata, On the solution of the V olterra integral equation of the second type for the error term in an asymptotic formula for arithmetic functions, Adv. Stud. Euro-Tbil. Math. J.15(2022), 83–92

2022
[2]

Kaczorowski and K

J. Kaczorowski and K. Wiertelak, Oscillations of the remainder term related to the Euler totient function, J. Number Theory130(2010), 2683–2700

2010
[3]

Kaczorowski and K

J. Kaczorowski and K. Wiertelak, Smoothing arithmetic error terms: the case of the Eulerφ-function, Math. Nachr.283(2010), no. 11, 1637–1645

2010
[4]

H. L. Montgomery, Fluctuations in the mean of Euler’s phi function, Proc. Indian Acad. Sci. Math. Sci.97(1987), no. 1–3, 239–245

1987
[5]

E. C. Titchmarsh, Introduction to the Fourier Integrals, 2nd ed., Clarendon Press, Oxford, 1948

1948
[6]

E. C. Titchmarsh, The Theory of the Riemann Zeta-function, 2nd ed., Clarendon Press, Oxford, 1986. NationalInstitute ofTechnology, GunmaCollege, Japan. Email address:iwata@gunma-ct.ac.jp

1986

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Iwata, On the solution of the V olterra integral equation of the second type for the error term in an asymptotic formula for arithmetic functions, Adv

H. Iwata, On the solution of the V olterra integral equation of the second type for the error term in an asymptotic formula for arithmetic functions, Adv. Stud. Euro-Tbil. Math. J.15(2022), 83–92

2022

[2] [2]

Kaczorowski and K

J. Kaczorowski and K. Wiertelak, Oscillations of the remainder term related to the Euler totient function, J. Number Theory130(2010), 2683–2700

2010

[3] [3]

Kaczorowski and K

J. Kaczorowski and K. Wiertelak, Smoothing arithmetic error terms: the case of the Eulerφ-function, Math. Nachr.283(2010), no. 11, 1637–1645

2010

[4] [4]

H. L. Montgomery, Fluctuations in the mean of Euler’s phi function, Proc. Indian Acad. Sci. Math. Sci.97(1987), no. 1–3, 239–245

1987

[5] [5]

E. C. Titchmarsh, Introduction to the Fourier Integrals, 2nd ed., Clarendon Press, Oxford, 1948

1948

[6] [6]

E. C. Titchmarsh, The Theory of the Riemann Zeta-function, 2nd ed., Clarendon Press, Oxford, 1986. NationalInstitute ofTechnology, GunmaCollege, Japan. Email address:iwata@gunma-ct.ac.jp

1986