Maximal order for divisor functions and zeros of the Riemann zeta-function

Hirotaka Akatsuka

Maximal order for divisor functions and zeros of the Riemann zeta-function

T0 review · 0 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The Riemann hypothesis holds exactly when one divisor function is bounded.

desk verdict A genuine new converse direction in an RH equivalence, with the load-bearing Omega-lemma surviving inspection; worth refereeing carefully. read the letter →

arxiv 2411.19259 v1 pith:EFS73UID submitted 2024-11-28 math.NT

Hirotaka Akatsuka This is my paper · ORCID

classification math.NT MSC 11N5611M26

keywords divisorfunctionsmaximalorderRiemannzeta-functionhypothesiszero-freeregionsuperiorhighlycompositenumberspartialEulerproductexplicitformula

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

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 $n$.

What carries the argument

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 $n$.

What would settle it

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).

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

Referee Report

0 major / 3 minor

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.

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.

minor comments (3)

[Abstract and Section 1] 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.
[Theorem 1 and Corollary 2] 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 8, Lemma 8.1] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorem 1's RH-equivalence is proved in both directions by self-contained explicit formulas and a Landau Omega-lemma; the only author self-citations are either re-derived in the text or non-load-bearing.

full rationale

The central claim (Theorem 1, Corollary 2) is derived both ways with no step assuming the conclusion. Direction (i)=>=(iii) uses the explicit formula (4.10) of Proposition 4.4, the SHCN asymptotics of Lemma 5.1, and the bound Z(kappa;X)=O(X^{Theta-kappa}/log X) from Lemma 4.5, which is unconditional and vanishes when Theta<=kappa; the limiting constant -zeta(kappa)B_kappa comes from the unconditional term log(-zeta(kappa)) in (4.10), not from any fitted parameter. Direction (ii)=>(i) is proved contrapositively in section 7: assuming a zero rho with Re rho>kappa, Proposition 7.1 gives an Omega_+ estimate through Landau's theorem (Lemma 7.2), whose pivot is Lemma 7.4(2). The load-bearing singularity was checked: near rho the transform ~F_+(s) has log(1/(sigma-beta)) coefficient kappa m_rho/(rho(rho-kappa)) != 0, so the singularity is non-removable and the contradiction in the proof of Proposition 7.1 stands. Self-citations to the author's [1] are not load-bearing: section 4 explicitly says 'many of the results in [1] are conditional. In order to discuss the behavior of the Euler product unconditionally as far as possible, we treat it again by using other methods' and re-derives the needed identities (Lemmas 4.1-4.4); Remark 1.1 and the proof of (i)=>(iii) of Theorem 3 cite [1] but add 'We give another proof here.' Lemma 8.1 cites [1, Lemma 2.1], but the identical identity is re-derived inside this paper via Lemma 4.1, Lemma 4.2, and eq. (5.4) in the proof of Lemma 5.1, so the citation is convenient rather than load-bearing. The normalization exp[li((log n)^{1-kappa})] is fixed by the classical Gronwall/Wigert leading order, and the resulting limsup value is a derived constant, not an input. No fitted inputs, no uniqueness claims imported from the authors, and no renamed known results were found. Score 1 reflects only the presence of minor, non-load-bearing self-citations; the derivation chain is otherwise self-contained.

Assumptions & free parameters 0 free parameters · 4 assumptions · 3 invented entities

The paper introduces no free parameters fitted to data. The limiting constants are derived from Mertens' theorem and the explicit formula. The main external axioms are standard results in analytic number theory: the explicit formula for zeta'/zeta, Landau's theorem, and the convergence of the zero series. The only new objects are definitions (a_kappa, E1, kappa-SHCNs), which are fully specified and do not constitute unverifiable postulates.

assumptions (4)

standard math Classical explicit formula for zeta'/zeta (Proposition 4.3) with sums over nontrivial zeros and trivial zeros.
Proven in Section 4 by residue calculus; used throughout for the partial Euler product expansion.
standard math Landau's theorem for Dirichlet series (Lemma 7.2).
Cited from Montgomery-Vaughan and used in Section 7 to detect zeros from non-negativity of an integral transform.
standard math Mertens' theorem and the prime number theorem with standard error terms.
Used in Lemmas 4.2, 5.1 and 6.2 to evaluate constants and error terms.
standard math Absolute convergence of sums over nontrivial zeros (sum over |gamma|^{-2} < infinity) to justify interchanges.
Invoked in Proposition 4.4 and Lemma 4.5.

invented entities (3)

kappa-superior highly composite numbers (kappa-SHCNs)
purpose: Define a sequence N_j maximizing sigma_kappa(n) for a given epsilon, linking large divisor values to the partial Euler product of zeta.
New definition in Section 2; a mathematical construction rather than a physical postulate. It is fully defined and not independently falsifiable.
Normalized divisor function a_kappa(n)
purpose: Normalize sigma_kappa(n) for the maximal-order statement in Theorem 1 and Corollary 2.
Defined in equation (1.4); an explicit function of standard arithmetic objects, not a postulated independent quantity.
Normalized partial Euler product E1(X)
purpose: Express a new equivalent condition for RH in terms of primes up to X.
Defined in equation (1.6); it is a normalization of the partial Euler product and is explicitly given.

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Pith. "Pith review of Maximal order for divisor functions and zeros of the Riemann zeta-function." pith.science (2026). https://pith.science/paper/EFS73UID

@misc{pith2026241119259,
  author       = {Pith},
  title        = {Pith review of: Maximal order for divisor functions and zeros of the Riemann zeta-function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EFS73UID}},
  note         = {Machine review of arXiv:2411.19259}
}

abstract

Ramanujan investigated maximal order for the number of divisors function by introducing some notion such as (superior) highly composite numbers. He also studied maximal order for other arithmetic functions including the sum of powers of divisors function. In this paper we relate zero-free regions for the Riemann zeta-function to maximal order for the sum of powers of divisors function. In particular, we give equivalent conditions for the Riemann hypothesis in terms of the sum of the $1/2$-th powers of divisors function. As a by-product, we also give equivalent conditions for the Riemann hypothesis in terms of the partial Euler product for the Riemann zeta-function.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

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

[1] [1]

Akatsuka, The Euler product for the Riemann zeta-func tion in the critical strip, Kodai Math

H. Akatsuka, The Euler product for the Riemann zeta-func tion in the critical strip, Kodai Math. J. 40 (2017) 79–101

work page 2017

[2] [2]

Alaoglu and P

L. Alaoglu and P. Erd˝ os, On highly composite and similar numbers, Trans. Amer. Math. Soc. 56 (1944) 448–469

work page 1944

[3] [3]

G. E. Andrews and B. C. Berndt, Ramanujan’s lost notebook . Part III, Springer, New York, 2012

work page 2012

[4] [4]

Broughan, Equivalents of the Riemann hypothesis

K. Broughan, Equivalents of the Riemann hypothesis. Vol . 1, Encyclopedia Math. Appl. 164, Cambridge University Press, Cambridge, 2017

work page 2017

[5] [5]

Broughan, Equivalents of the Riemann hypothesis

K. Broughan, Equivalents of the Riemann hypothesis. Vol . 3, Encyclopedia Math. Appl. 187, Cambridge University Press, Cambridge, 2024

work page 2024

[6] [6]

Caveney, J.-L

G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA n umbers, Ramanujan J. 29 (2012) 359–384

work page 2012

[7] [7]

Erd˝ os and J.-L

P. Erd˝ os and J.-L. Nicolas, R´ epartition des nombres su perabondants, Bull. Soc. Math. France 103 (1975) 65–90

work page 1975

[8] [8]

T. H. Gronwall, Some asymptotic expressions in the theor y of numbers, Trans. Amer. Math. Soc. 14 (1913) 113–122

work page 1913