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REVIEW 2 major objections 4 minor 6 references

Analytic expressions pertaining to certain arithmetical functions

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that the sum-of-divisors function has an explicit infinite-series representation, which converts the Riemann hypothesis into a concrete inequality holding for every real parameter $t>0$.

desk verdict The inversion method is worth a look, but the sigma(N) formula and the RH criterion collapse on a factor-R^2 algebra error. read the letter →

arxiv 2411.17327 v1 pith:DMJOSIA7 submitted 2024-11-26 math.CO math.NT

Aung Phone Maw This is my paper · ORCID
classification math.COmath.NT MSC 11A2511D0911M26
keywords analyticrepresentationarithmeticfunctionsDiophantineequationssumofdivisorsRiemannhypothesisRobin-LagariascriterionMittag-Lefflerexpansionpartialfractioninversion
topics The Riemann Hypothesis
open problems 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 develops a method for writing certain arithmetical sums as convergent infinite series, by inverting a formal partial-fraction series through a Mittag-Leffler expansion. It applies the method to sums over positive integer solutions of $kb^2+da^2=N$ and $kb^2-da^2=N$, and then to the sum-of-divisors function $\sigma(N)$. The central consequence is Proposition 3: the Riemann hypothesis is equivalent to the inequality (1.13) holding for all $N>1$ for any fixed real $t>0$, because the left side is claimed to equal $\sigma(N)$. If the representation is exact, the Lagarias criterion becomes a one-parameter family of elementary-looking inequalities. A sympathetic reader should care because this would be a new analytic handle on a classical arithmetic function and on the Riemann hypothesis.

What carries the argument

The engine is an inversion theorem (Theorem 5) for the formal series $\sum_{n\ge 1} f(n)/(n+z)$: if this series equals $F(z)$, then $f(N)$ is recovered as an infinite series of differences $F(k-N\pm it)-F(k-N\mp it)$ weighted by integrals $\int_0^1 \cos(\pi k\beta)/\cosh(\pi\beta t)\,d\beta$. Applied to $F(z)$ from Lemma 6, this yields explicit representations for the square-indicator functions $q_{k,s}(N)$, hence, after summing over $a$, for the divisor sums in Propositions 15 and 16, and finally for $\sigma(N)$ in Proposition 18.

What would settle it

Evaluate the right-hand side of Proposition 18 for a small integer such as $N=2$ at a fixed real $t>0$, truncating the convergent series, and compare with $\sigma(2)=3$; if the value does not approach $\sigma(2)$ as more terms are included, the claimed representation fails. Alternatively, check the denominator exponent in the first term of Eq. (4.7): if it is genuinely $(N+c)$, then Propositions 18 and 3 cannot follow from the stated substitution.

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Extended reading notes

Core claim

The paper's central claim is that the arithmetic function $\sigma(N)$ admits the explicit series representation (6.2), obtained by substituting the analytic expression for $q_1(4N+a^2)/(4N+a^2)^2$ into the divisor-sum identity $\sigma(N)=q_1(N)\sqrt{N}+\sum_{a=1}^{N-1} q_1(4N+a^2)\sqrt{4N+a^2}$. Because the left side of (6.2) is asserted to equal $\sigma(N)$ for every real $t>0$, the Robin-Lagarias inequality $\sigma(N)<H_N+e^{H_N}\log H_N$ is claimed equivalent to the inequality (1.13) for any fixed $t>0$. The paper thereby presents proving or disproving the Riemann hypothesis as proving or disproving a single concrete inequality that does not refer to the zeta function.

Load-bearing premise

The central claim depends on the substitution step from Eq. (4.7) into identity (6.1) being algebraically exact, and the printed equations show a denominator mismatch ($N+c$ versus $(N+c)^2$) in the first term that would break that exactness if it is not a typographical error.

Editorial extensions

If this is right

  • If Proposition 18 holds, $\sigma(N)$ is given by a convergent infinite series involving hyperbolic functions and the auxiliary quantities $G_{N,t}$.
  • The Riemann hypothesis becomes equivalent to the single inequality (1.13) for any fixed real $t>0$.
  • The Robin and Lagarias bounds become accessible through term-by-term estimates of the series representation.
  • The method extends to sums with $b^{4s}$ and more general expressions $y(a)$ in the Diophantine equation, as the paper remarks after Proposition 16.

Reading between the lines

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

  • The free parameter $t$ creates a family of equivalent formulations of the Riemann hypothesis; one could try to choose $t$ to make numerical verification of the inequality easier.
  • If the series representation for $\sigma(N)$ is genuinely convergent and uniform in $N$, it might connect divisor sums to special values of hyperbolic and trigonometric sums, opening new comparisons with known estimates.
  • The denominator mismatch noted in the weakest-assumption field is directly testable by machine computation before any serious appeal to the Riemann hypothesis is made.
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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

2 major / 4 minor

Summary. The paper proposes a general method, based on a Fourier-inversion lemma for series of partial fractions, to derive analytic expressions for sums over positive integer solutions of Diophantine equations of the forms k b^2 + d a^2 = N and k b^2 - d a^2 = N. It applies the method to the arithmetic functions q_{k,s}(N), obtains explicit infinite-series formulas for certain divisor-type sums, and finally claims a new equivalent form of the Robin-Lagarias criterion for the Riemann hypothesis by writing sigma(N) as an infinite series with a free parameter t.

Significance. If the main identities were correct, the paper would provide striking explicit analytic representations for arithmetic sums that are normally accessible only through modular or sieve-theoretic methods, and the parametrized reformulation of the Lagarias criterion would be a new equivalent statement of the Riemann hypothesis. The method itself, particularly the inversion theorem in Section 2, is potentially of independent interest. However, the central sigma(N) identity contains a concrete algebra error in the leading term, so the main claim is not established. The Riemann hypothesis part is, in any case, a direct restatement of Lagarias' theorem once sigma(N) is identified with the proposed series; it supplies no independent evidence about the Riemann hypothesis. The paper also provides no numerical verification of any of its long identities, which is a serious gap in view of the complexity of the formulas.

major comments (2)
  1. [Section 6, Eqs. (4.7), (6.1), (6.2)] Proposition 18 does not follow from the stated substitution. In Eq. (6.1), the term for sigma(N) is q1(4N+a^2)/(4N+a^2)^2 multiplied by (4N+a^2)^{5/2}. Substituting Eq. (4.7) with k=1, c=a^2, and N replaced by 4N gives, for R=4N+a^2, the leading contribution pi^2/(3R(e^{2 pi t}-1)) times R^{5/2}, i.e. pi^2 R^{3/2}/(3(e^{2 pi t}-1)). The first term printed in Eq. (6.2) is instead pi^2/(3(e^{2 pi t}-1) R^{1/2}), which differs from the substituted value by a factor R^2 = (4N+a^2)^2. Thus Eq. (6.2) is not the result of substituting Eq. (4.7) into Eq. (6.1), and Proposition 18 is unproved as stated.
  2. [Section 6, Proposition 3 and Eq. (6.6)] Since the inequality in Proposition 3, Eq. (1.13), is exactly the left-hand side of Eq. (6.6), and since Eq. (6.6) is obtained from the incorrect Proposition 18, the claimed equivalence with the Riemann hypothesis is not supported. The paper does not establish that the left side of (1.13) equals sigma(N), so the parametrized inequality is not a proved equivalent of Lagarias' criterion. This is a load-bearing failure: the central advertised consequence of the paper depends on the invalid substitution.
minor comments (4)
  1. [Throughout] There are numerous typographical and OCR-style defects in the long displayed formulas, such as unbalanced parentheses in the definition of G_{M,t,k} in Eq. (1.8) and inconsistent exponents in several places; these make independent verification substantially harder.
  2. [Section 1 and Section 6] The paper describes Proposition 3 as a 'possible improvement' of the Robin-Lagarias criteria, but the free parameter t enters only through an identity that is claimed to hold for all t>0; without the sigma(N) identity the statement is merely a restatement of Lagarias' theorem, and with it it is an equivalent reformulation rather than an improvement.
  3. [Section 5, Eqs. (5.11)-(5.13)] The star convention for singular summands is introduced informally and used in the derivation of Propositions 14-16, but the equivalence between the original finite sum and the star-modified infinite series is not proved. This is not directly involved in Proposition 18, but it is a gap in the earlier derivations.
  4. [Section 4, Lemma 12] The closed forms in Lemma 12 are stated after only a one-line indication of proof; since later substitutions in Sections 5 and 6 depend on exact constants, a fuller derivation or numerical cross-check for these constants would be needed.

Circularity Check

1 steps flagged · score 2.0 of 10

No substantive circularity in the analytic derivations; the only mild issue is that Proposition 3 restates Lagarias's criterion by substitution rather than providing an independent RH argument.

  1. renaming known result [Section 6, Proposition 3 and Eq. (6.6), page 24-25]
    "Now we use or representation of σ(N ) and substitute in the inequality to get the following equivalent of (6.5). ... Since the expression on the left side of the inequality always takes the value σ(N ) for all real t > 0. Proving or disproving the inequality (6.6), for any choice of real value t > 0 amounts to proving or disproving the Riemann hypothesis, thus leading to the statement of Proposition 3."

    The 'new' Riemann hypothesis criterion is not an independent derivation: Eq. (6.6) is obtained by substituting the σ(N) representation into Lagarias's inequality (6.5), so its equivalence to RH is inherited from Lagarias's theorem. The left side is defined to equal σ(N) through Proposition 18, making the inequality a restatement of (6.5) in the new series coordinates. This does not undermine the independent derivation of the σ-series itself, but Proposition 3 adds no new RH evidence beyond Lagarias's criterion.

full rationale

The main derivation chain is self-contained: q_{k,s}(N) is defined independently in (3.1), Lemma 6 computes the generating function from ζ-values and the Mittag-Leffler expansion, Theorem 5 is an inversion of the partial-fraction series, and Propositions 8, 14, 15, 16, 17 and 18 follow by substitution of previously derived identities. No fitted constants occur, no parameter is tuned to a subset of data, and no load-bearing self-citation appears; the cited external results (Ramanujan's expansion, Robin, Lagarias) are standard and are not used to force the conclusion. The Riemann hypothesis equivalence in Proposition 3 is an explicit substitution of the derived σ(N) series into Lagarias's inequality, so it is a reformulation rather than a circular inference. The algebraic mismatch in the leading term of Proposition 18 noted in the reader's take is a correctness concern about the substitution from (4.7), not a circularity in the derivation structure. Overall, the paper's analytic identities are not equivalent to their inputs by construction, and the only mild circularity flavor is the presentation of Lagarias's criterion under a new series form.

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

The derivation relies on standard Fourier and Mittag-Leffler material plus Robin-Lagarias as external input. There are no fitted constants; the only free auxiliary parameter is t. The star convention is an ad hoc regularization that is not fully justified.

free parameters (1)
  • t = arbitrary, t > 0
    Auxiliary real parameter appearing in all identities and in the Riemann hypothesis inequality. The paper claims every t > 0 works, so it is not fitted to data, but it is an unconstrained parameter that appears throughout.
assumptions (4)
  • standard math Mittag-Leffler expansion (1.2) for the hyperbolic kernel
    Used throughout as the starting identity; attributed to Ramanujan's notebooks via Berndt [1].
  • standard math Fourier coefficient recovery on [-1,1] used in Theorem 5
    Used to extract f(N) from the trigonometric series; relies on uniform convergence of the intervening series.
  • domain assumption Robin's and Lagarias's theorems equating the Riemann hypothesis with divisor-sum inequalities
    Basis for Proposition 3; accepted from [6] and [2] as external results.
  • ad hoc to paper Star convention: singular summands are set to zero while the epsilon_N term is retained
    Introduced in Propositions 1, 14 and 15; the paper does not prove that this regularization reproduces the original finite sums over positive integer solutions.

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Cite this review

Pith. "Pith review of Analytic expressions pertaining to certain arithmetical functions." pith.science (2026). https://pith.science/paper/DMJOSIA7

@misc{pith2026241117327,
  author       = {Pith},
  title        = {Pith review of: Analytic expressions pertaining to certain arithmetical functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DMJOSIA7}},
  note         = {Machine review of arXiv:2411.17327}
}
abstract

We demonstrate the general outlines of a method for obtaining analytic expressions for certain types of general arithmetical sums. In particular, analytical expressions for a general arithmetical sum whose terms are summed over either the positive integer solutions $(a,b)$ of the Diophantine equation $kb^2+da^2 = N$ or the positive integer solutions $(a,b)$ of the Diophantine equation $kb^2-da^2 = N$ are derived. As one of the consequences, we propose a possible improvement of the Robin-Lagarias criteria for the Riemann hypothesis.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 5 canonical work pages

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    Bruce C. Berndt. Ramanujan’s Notebooks Part II. Springe r New York, NY, 1988. url: https://doi. org/10.1007/978-1-4612-4530-8

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    An Elementary Problem Equivalent t o the Riemann Hypothesis

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    work page doi:10.2307/2695443 2002

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    On the Partition Function p(n)

    H. A. Rademacher. “On the Partition Function p(n)”. In: P roceedings of The London Mathematical Society (1938), pp. 241–254. url: https://api.semanticscholar.o rg/CorpusID:120061968

    work page 1938

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    Theory of Complex Functions

    Reinhold Remmert. Theory of Complex Functions. Springe r New York, NY, 2012. url: https://doi. org/10.1007/978-1-4612-0939-3

    doi:10.1007/978-1-4612-0939-3 2012

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    Ueber die Anzahl der Primzahlen unter e iner gegebenen Gr¨ osse

    G. F. B. Riemann. “Ueber die Anzahl der Primzahlen unter e iner gegebenen Gr¨ osse”. In: Monatsberichte der K/dieresis.ts1oniglich Preußischen Akademie der Wissenschaften zu Berli n (1859), pp. 671–680

    work page

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    Grandes valeurs de la fonction somme des divis eurs et hypoth` ese de Riemann

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    work page 1984

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Reviewed August 12, 2026 · model on record in the stance chip above.