CA2 numbers and the Riemann hypothesis: a numerical investigation of Ramanujan's inequalities

Dmitry R. Leshchiner; James Thompson; Oleg R. Musin

REVIEW

CA2 numbers and the Riemann hypothesis: a numerical investigation of Ramanujan's inequalities

Dmitry R. Leshchiner, Oleg R. Musin, James Thompson

Number Theory · Submitted 29 Oct 2026 · 12 references

Pith's verdict: Section 2.2 is a real, checkable derivation (Briggs's slope from Ramanujan's bounds); the numerical drift evidence is honest but rests on unreleased data and a borderline zero-sum truncation.

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CitationDmitry R. Leshchiner, James Thompson, Oleg R. Musin. "CA2 numbers and the Riemann hypothesis: a numerical investigation of Ramanujan's inequalities." arXiv:2610.39120 (2026). https://pith.science/paper/ASYOZ6P4

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Editor's note

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The short version: one clean mathematical observation, plus an honest numerical half that cannot currently be audited. Section 2.2 is the part worth keeping. Starting from Ramanujan's conditional bounds (2.4), the authors show in a line that log Δ(n) = −(1/2) log log n + O(1) along CA numbers under RH, which is stronger than Briggs's empirically observed slope and explains it as a forced consequence rather than a coincidence. I checked the step; it is correct given (2.4), and (2.4) is quoted from published sources, so importing it is fine. I have not seen this derivation stated elsewhere. The record-set reduction B in (3.1) is also valid, and the paper honestly notes it only helps if records can be located efficiently.

The numerical half is where I hesitate, and the authors are upfront about the reasons. Section 4.3 says the point tables and programs are not included, the figures have no error bars, and the quoted ranges are approximate. With K = 1000 zeta ordinates, the tail bound for Q at u ≈ 25 is about 0.036. The global drift in Q (roughly 0.36 to 1.25) is an order of magnitude larger, so the headline trend survives that bound. But the local oscillations the paper asks you to look at are comparable to the missing tail, the truncation error grows with u, and the bound itself is crude since each cosine is set to one. The fine structure is therefore not established by the released material. This is a limitation the authors acknowledge, not a hidden flaw, but it is load-bearing for Questions 3 and 4.

Minor points: Wu's theorem is imported from an arXiv preprint with no independent verification here, and one of the two anchors for (2.4) is the authors' own paper. Neither disqualifies the work, and both deserve referee attention. The paper also correctly says that Wu's reduction does not transfer to Q, so the CA2-versus-CA residual comparison is an open conjecture, not a consequence of the theorem.

Who this is for: people working on Robin's criterion, CA numbers, and sharp constants for σ(n). Section 2.2 deserves citation; the numerics deserve a companion with released data and a K = 10000 comparison. I would send this to a serious referee. If I were the editor, the acceptance would be conditional on releasing the point tables and either raising K or scoping the claims to what the evidence supports.

What would settle it

Extend the CA computation — the paper generates transition factors through CA index $10^{10}$ and evaluates $T$ through index $24\cdot 10^6$ — and test (2.4) asymptotically: if the empirical liminf of $T$ along CA numbers falls below $c_1\approx 1.3932$ or the limsup climbs above $c_2\approx 1.5578$ on arbitrarily long runs, the imported Ramanujan bounds fail and equation (2.7) fails with them; isolated out-of-range points would not settle it, since (2.4) is asymptotic. A directly targeted check is whether $\log\Delta(n)+\frac{1}{2}\log\log n$ stays bounded as $u$ approaches 25 and beyond — unbounded drift in that quantity refutes the $O(1)$ remainder by itself. Repeating the residual computation with $K=10^4$ or $K=10^5$ zeros instead of $K=1000$ would show whether the drift of $Q$ is partly an artifact of truncation error, which the paper estimates at about $0.036$ for $Q$ near $u=25$.

Referee report

Significance

If the formal part is accepted, the paper's main theorem (2.7) is correct and genuinely useful: it shows that Briggs's -1/2 slope is not an empirical coincidence but a consequence of Ramanujan's conditional bounds under RH. The record-set reduction in Section 3.2 is also sound. A further strength is that the paper does not fit parameters to manufacture agreement: the proposed Q <= 3/2 bound is explicitly left as an open question, and the authors state their truncation scales and assumptions, including the RH-dependence of R(n). The numerical contribution, however, is not yet auditable: the central drift and CA2-resemblance claims rest on plotted curves with no released point tables, no error bars, and only hand-read numerical ranges, while the K=1000 truncation error is not negligible at the per-window scale used for Q. These issues are fixable within the paper's scope, so I do not regard them as grounds for rejection, but they require substantive revision.

Major objections (3)

[Section 4.3, Figures 3-6] The central experimental observations--the slowly varying drift in D and Q, the CA2 analogue in Figure 5, and the consistency with Q <= 3/2--are supported only by plotted curves. Section 4.3 states that the underlying point tables and programs are not included, and Section 3.3 reports the key numerical ranges as values 'read approximately from the displayed curves.' Because these observations motivate Questions 3-4 and the proposed bound in (3.2), they are load-bearing for the numerical half of the paper. I request that the authors release the point tables, or at least machine-readable per-window minimum, maximum, and count statistics for both the CA and CA2 samples, so that the reported drift and the CA2 resemblance can be verified independently of the figures. The authors acknowledge this: “The numerical ranges quoted from the figures are approximate.”
[Section 4.2, Eq. (4.1), Table 1] With K=1000, the worst-case truncation error in Q is uE_K, which at u=25 is about 0.036. This is the same order as the small local oscillations in Figure 4 and is large relative to the per-window movements used in Section 3.3 to discuss whether Q is stabilizing or drifting. The authors acknowledge that resolving stabilization requires a tail scale below the measured changes, but the displayed computations nevertheless use K=1000. Please rerun the residual computations with K=10,000 or 100,000, or otherwise display the tail bound as an error band on the Q plots, so that the observed drift and the consistency with Q <= 3/2 can be distinguished from a fixed truncation artifact. The authors acknowledge this: “Resolving stabilization, or the higher-order coefficients, requires a tail scale below the changes being measured, together with control of arithmetic errors.”
[Section 3.4, Figure 5, Question 5] The claim that the CA2 residual resembles the full-CA residual is made by visual comparison of samples with very different spacing. Section 3.4 itself states that a direct test 'would compare both samples in identical u-windows and record their minimum, maximum, and number of points,' but that test is not performed. Since Question 5 and the proposed use of CA2 as a cheaper proxy for the residual depend on this resemblance, the paper should include the window-by-window statistics it recommends; without them, the CA2 observations are not auditable evidence. The authors acknowledge this: “The existing curves suggest the com- parison, but do not supply those window statistics. 3.5 Local behavior of the residual For the residual, the class pattern is more complicated.”

Minor objections (4)

[Section 2.1, Eq. (2.4)] The symbols R_L and R_U for the lower and upper limits of T are visually too close to the function R(n); consider renaming them, for example L_T and U_T, to avoid confusion.
[Section 5.1, Eq. (5.2)] The displayed expansion is easy to misread; write it as P(n) = A_1/u + A_2/u^2 + ... + A_r/u^r + o(u^{-r}) to make the powers of u unambiguous.
[Section 3.3, Figure 4 caption] The caption states 'uD = e^gamma Q'; writing Q = e^{-gamma} uD in the caption would make the normalization in the text easier to follow.
[References] Reference [12] is cited as a 2019 preprint; if a published version now exists, it should be cited in its final form.

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The paper

The paper claims

Under RH, the paper derives log Δ(n) = −1/2 log log n + O(1) along colossally abundant numbers from Ramanujan's conditional bounds, explaining Briggs's observed slope, then probes the next term against a sum over zeta zeros.

The reading

Robin's criterion ties the Riemann hypothesis to a single inequality involving the sum of divisors, and the inequality need only be checked on the colossally abundant (CA) integers, which maximize $\sigma(n)/n^{1+\epsilon}$; Wu's theorem narrows that check to the sparse subclass CA2. The paper's main result is that, under RH, Ramanujan's conditional bounds force the decay law $\log\Delta(n) = -\frac{1}{2}\log\log n + O(1)$ along CA numbers, where $\Delta(n)$ is the gap between $e^\gamma\log\log n$ and $\sigma(n)/n$; this turns the slope seen in Briggs's computations into a theorem with a bounded remainder rather than a fitted trend. The numerical half of the paper compares the normalized gap $T(n)$ with Ramanujan's oscillatory expression $R(n)$ built from the nontrivial zeta zeros: the residual is small and slowly varying, while the scaled residual $Q(n)=e^{-\gamma}uD(n)$ keeps drifting across the sampled range without stabilizing, so the next coefficient remains unmeasured. The paper also shows that testing Robin's inequality only on the record values of $G(n)=\sigma(n)/(n\log\log n)$ on CA2 preserves the equivalence to RH, and it sets out open questions on convergence, an eventual upper bound $Q\le 3/2$, and whether CA2 reproduces the full CA residual envelope.

What carries the argument

The load-bearing object is the normalized Robin gap $T(n)=\left(e^\gamma\log\log n-\sigma(n)/n\right)\sqrt{\log n}$ together with Ramanujan's two-sided bounds (2.4): along the CA numbers, $c_1\le\liminf T\le\limsup T\le c_2$ with $c_1\approx 1.3932$ and $c_2\approx 1.5578$. Boundedness of $T$ above and away from zero is exactly the mechanism that converts the multiplicative estimate for $\sigma(n)/n$ into the additive asymptotic law $\log\Delta(n)=-\frac{1}{2}\log\log n+O(1)$, because $\Delta(n)=T(n)/\sqrt{\log n}$ and $\log T(n)=O(1)$. For the next term the machinery is the convergent sum over zeros $R(n)=e^\gamma\left(2(\sqrt{2}-1)-2\sum_{j\ge1}\frac{\cos(t_j u)}{1/4+t_j^2}\right)$, which under RH lies between $c_1$ and $c_2$ and supplies Ramanujan's approximation to $T$; subtracting it defines the residuals $D$, $P=e^{-\gamma}D$, and $Q=uP$ displayed in the figures. The reduction machinery is Wu's theorem — RH holds if and only if Robin's inequality holds on CA2 above 5040 — which legitimizes the far sparser CA2 sample, together with the critical-parameter construction (2.8) that generates the CA sequence; the paper notes that Wu's reduction transfers to $G$ but not automatically to the residual $Q$.

Extended reading notes

Core claim

The paper's central claim is that the logarithmic decay law Keith Briggs observed numerically — $\log\Delta(n)\approx -\frac{1}{2}\log\log n$ along colossally abundant numbers, with $\Delta(n)=e^\gamma\log\log n-\sigma(n)/n$ — is a forced consequence of Ramanujan's conditional inequalities, not an empirical coincidence. Using the normalized gap $T(n)=\Delta(n)\sqrt{\log n}$, the paper recalls Ramanujan's bounds (2.4), which put the liminf and limsup of $T$ along CA numbers between $c_1\approx 1.3932$ and $c_2\approx 1.5578$; since $\Delta(n)=T(n)/\sqrt{\log n}$ and $\log T(n)=O(1)$, this yields $\log\Delta(n)=-\frac{1}{2}\log\log n+O(1)$ (equation 2.7), with a bounded remainder that is stronger than a leading-slope statement. The bounds themselves come from Ramanujan's posthumously published notes and from related work the paper cites; the paper's own contribution is the derivation of the log law, the record-value reduction of the test set, and the numerical comparison at the next order. In that comparison, $T(n)$ is measured against the oscillatory sum over zeta zeros $R(n)=e^\gamma\left(2(\sqrt{2}-1)-2\sum_{j\ge 1}\frac{\cos(t_j u)}{1/4+t_j^2}\right)$: the residual $D=R-T$ is small and slowly varying on both CA and CA2, while $Q(n)=e^{-\gamma}uD(n)$ rises from roughly $0.36$ to $1.25$ over the sampled range without settling. The paper states explicitly that the observations do not determine a limiting coefficient, and leaves the convergence of $Q$, the proposed bound $Q\le 3/2$, the equality of CA and CA2 limits, and the existence of a constant-coefficient asymptotic expansion as open questions.

Load-bearing premise

The derivation of the log law rests entirely on Ramanujan's two-sided bounds (2.4) — that along CA numbers the normalized gap $T(n)$ is eventually bounded away from zero by $c_1$ and eventually bounded above by $c_2$ — which the paper imports from [10] and [4] rather than proving; if either bound fails on the CA sequence, the bounded remainder in $\log\Delta(n)=-\frac{1}{2}\log\log n+O(1)$ collapses, and the numerical comparison with $R(n)$ presumes RH, as the authors acknowledge.

Editorial extensions

If this is right

Briggs's observed slope is explained rather than fitted: under RH, any computation along CA numbers must show $\log\Delta(n)=-\frac{1}{2}\log\log n+O(1)$, so a materially different fitted slope would signal numerical error or a failure of Ramanujan's bounds.
The record-value set $B$ of successive maxima of $G$ on CA2 is itself sufficient: Robin's inequality on $B$ is equivalent to RH, so the test set can be thinned below Wu's CA2 while preserving the reduction.
Ramanujan's zero sum $R(n)$ tracks $T(n)$ closely on both CA and CA2 — the residual is small and slowly varying — supporting the view that the nontrivial zeta zeros shape $\sigma(n)/n$ at the next order of accuracy.
If the eventual bound $Q(n)\le 3/2$ holds, then $T(n)\ge R(n)-\frac{3e^\gamma}{2\log\log n}$ eventually along CA numbers, turning Ramanujan's one-sided estimate into a two-sided effective inequality; the plotted data are consistent with the bound but do not determine a threshold.
Because $Q$ does not stabilize over $14\lesssim u\lesssim 25$, the next coefficient in the proposed $1/u$ expansion of $P(n)$ has not been measured; distinguishing convergence from persistent oscillation requires longer ranges and more retained zeros, with the tail error pushed below the changes being measured.

Reading between the lines

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

If the paper's equation (2.7) is right, the $-1/2$ slope becomes a cheap consistency check for any computational CA generator: confirming the bounded remainder is a diagnostic independent of the subtler residual analysis. The paper does not propose this use.
The record-value reduction raises a sharper version of Question 2: can membership in $B$ be decided without evaluating $G$ on every CA2 point, and do iterated record rules produce test sets of density zero within CA2 whose union would still catch any RH counterexample?
Wu's open multiplier-2 case (Question 1) is directly testable from the paper's own transition data: a single CA3 step with multiplier 2 whose $G$ increment is negative would answer Wu's question negatively.
The drift of $Q$ from about 0.36 to 1.25 across one decade of $u$ is too short to distinguish a slow divergence from a long oscillation; the discriminating experiment is windowed extrema of $Q$ on equal $u$-windows over both CA and CA2 — statistics the paper explicitly says its curves do not yet supply.

Original abstract

Robin's criterion reduces the Riemann hypothesis to an inequality for colossally abundant (CA) numbers, and Wu's theorem reduces the test further to the subclass CA2. We study this reduction numerically and consider smaller subsets on which the same criterion remains sufficient. We also show that the logarithmic decay law suggested by Briggs's computations follows from Ramanujan's conditional bounds. To investigate the next term, we compare the normalized Robin difference with Ramanujan's oscillatory expression involving the nontrivial zeros of the zeta function. The resulting difference has a slowly varying component on both CA and CA2. Its scaled values continue to vary over the computed range, leading to questions about convergence, upper bounds, and higher-order terms. We describe the sampling used in the figures and estimate the error from truncating the sum over zeros.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-30: 1 request, 5,476 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.81 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.73 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Robin's criterion reduces the Riemann hypothesis to an inequality for colossally abundant (CA) numbers, and Wu's theorem reduces the test further to the subclass CA2. We study this reduction numerically and consider smaller subsets on which the same criterion remains sufficient. We also show that the logarithmic decay law suggested by Briggs's computations follows from Ramanujan's conditional bounds. To investigate the next term, we compare the normalized Robin difference with Ramanujan's oscillatory expression involving the nontrivial zeros of the zeta function. The resulting difference has a slowly varying component on both CA and CA2. Its scaled values continue to vary over the computed range, leading to questions about convergence, upper bounds, and higher-order terms. We describe the sampling used in the figures and estimate the error from truncating the sum over zeros.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Robin's criterion reduces the Riemann hypothesis to an inequality for colossally abundant (CA) numbers, and Wu's theorem reduces the test further to the subclass CA2.	background 1.00	0.05	0.08	asserted 1.00
s1	We study this reduction numerically and consider smaller subsets on which the same criterion remains sufficient.	contribution 0.99	0.90	0.07	asserted 0.83
s2	We also show that the logarithmic decay law suggested by Briggs's computations follows from Ramanujan's conditional bounds.	result 0.95	0.88	0.19	asserted 1.00
s3	To investigate the next term, we compare the normalized Robin difference with Ramanujan's oscillatory expression involving the nontrivial zeros of the zeta function.	method 0.85	0.87	0.04	asserted 0.86
s4	The resulting difference has a slowly varying component on both CA and CA2.	result 1.00	0.87	0.12	asserted 1.00
s5	Its scaled values continue to vary over the computed range, leading to questions about convergence, upper bounds, and higher-order terms.	interpretation 0.65	0.86	0.13	asserted 0.90
s6	We describe the sampling used in the figures and estimate the error from truncating the sum over zeros.	method 0.99	0.93	0.03	asserted 0.98

What kind of paper the classifier reports this is: simulation study 0.90 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Overview of G(n) = σ(n)/(n log log n). The displayed values remain below e γ . On this scale the local oscillations are difficult to see and most points appear green (CA3)

**Figure 2.** Enlarged views of G. Blue CA1 points occur on descending portions, while orange/red CA2 points appear near changes of behavior. These plots resolve variations that are not visible in

**Figure 3.** The curves labeled T(n) and R(n) on the CA2 sample, with horizontal coordinate u = log log n. Their visible oscillations are similar, but the vertical separation changes across the range. The figure includes every point of our CA2 sample

**Figure 4.** Residual scales on sampled CA numbers. Left:

**Figure 5.** shows the positions of the CA2 sample and its residual. The points are sparse and unevenly spaced on the u axis. Nevertheless, the broad increase and the small oscillations of the residual resemble those of the entire CA sequence sampled in

**Figure 6.** Class-colored residual: overview and detail. The vertical axes show

Figures reproduced from arXiv:2610.39120 by the authors.

References

Works cited (12)

[1]

L. Alaoglu and P. Erdős, On highly composite and similar numbers,Trans. Amer. Math. Soc.56(1944), 448–469. doi:10.1090/S0002-9947-1944-0011087-2. 13

View on Pith doi:10.1090/s0002-9947-1944-0011087-2 1944
[2]

K. Briggs, Abundant numbers and the Riemann hypothesis,Experimental Mathe- matics15(2006), no. 2, 251–256. doi:10.1080/10586458.2006.10128957

View on Pith doi:10.1080/10586458.2006.10128957 2006
[3]

T. Morrill and D. J. Platt, Robin’s inequality for 20-free integers,Integers21(2021), A28. arXiv:1809.10813

View on Pith Pith review arXiv 2021
[4]

O. R. Musin, Ramanujan’s theorem and highest abundant numbers,Arnold Math. J. 6(2020), 119–130. doi:10.1007/s40598-020-00136-w

View on Pith doi:10.1007/s40598-020-00136-w 2020
[5]

O. R. Musin, Higher-order colossally abundant numbers, preprint (2026), arXiv:2609.33794

View on Pith Pith review arXiv 2026
[6]

J.-L. Nicolas, The sum of divisors function and the Riemann hypothesis,Ramanujan J.58(2022), no. 4, 1113–1157. doi:10.1007/s11139-021-00491-y

View on Pith doi:10.1007/s11139-021-00491-y 2022
[7]

J.-L. Nicolas and J. Sondow, Ramanujan, Robin, highly composite numbers, and the Riemann hypothesis, inRamanujan 125, Contemporary Mathematics, vol. 627, American Mathematical Society, 2014, 145–156. arXiv:1211.6944

View on Pith Pith review arXiv 2014
[8]

A. M. Odlyzko, Tables of zeros of the Riemann zeta function,https://www-users .cse.umn.edu/~odlyzko/zeta_tables/index.html

View on Pith
[9]

S. Ramanujan, Highly composite numbers,Proc. London Math. Soc.(2)14(1915), 347–409. doi:10.1112/plms/s2_14.1.347

View on Pith doi:10.1112/plms/s2_14.1.347 1915
[10]

S. Ramanujan, Highly composite numbers, annotated and with a foreword by J.- L. Nicolas and G. Robin,Ramanujan J.1(1997), 119–153; published as “Highly Composite Numbers by Srinivasa Ramanujan.” doi:10.1023/A:1009764017495

View on Pith doi:10.1023/a:1009764017495 1997
[11]

G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Rie- mann,J. Math. Pures Appl.(9)63(1984), 187–213

View on Pith 1984
[12]

X. Wu, Subsets of colossally abundant numbers, preprint (2019), arXiv:1903.03490. Dmitry R. Leshchiner email: leshch6@gmail.com Oleg R. Musin University of Texas Rio Grande Valley, Brownsville, TX 78520, USA. email: oleg.musin@utrgv.edu James Thompson email: james.thompson.5548@gmail.com 14

View on Pith Pith review arXiv 2019

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

L. Alaoglu and P. Erdős, On highly composite and similar numbers,Trans. Amer. Math. Soc.56(1944), 448–469. doi:10.1090/S0002-9947-1944-0011087-2. 13

View on Pith doi:10.1090/s0002-9947-1944-0011087-2 1944

[2] [2]

K. Briggs, Abundant numbers and the Riemann hypothesis,Experimental Mathe- matics15(2006), no. 2, 251–256. doi:10.1080/10586458.2006.10128957

View on Pith doi:10.1080/10586458.2006.10128957 2006

[3] [3]

T. Morrill and D. J. Platt, Robin’s inequality for 20-free integers,Integers21(2021), A28. arXiv:1809.10813

View on Pith Pith review arXiv 2021

[4] [4]

O. R. Musin, Ramanujan’s theorem and highest abundant numbers,Arnold Math. J. 6(2020), 119–130. doi:10.1007/s40598-020-00136-w

View on Pith doi:10.1007/s40598-020-00136-w 2020

[5] [5]

O. R. Musin, Higher-order colossally abundant numbers, preprint (2026), arXiv:2609.33794

View on Pith Pith review arXiv 2026

[6] [6]

J.-L. Nicolas, The sum of divisors function and the Riemann hypothesis,Ramanujan J.58(2022), no. 4, 1113–1157. doi:10.1007/s11139-021-00491-y

View on Pith doi:10.1007/s11139-021-00491-y 2022

[7] [7]

J.-L. Nicolas and J. Sondow, Ramanujan, Robin, highly composite numbers, and the Riemann hypothesis, inRamanujan 125, Contemporary Mathematics, vol. 627, American Mathematical Society, 2014, 145–156. arXiv:1211.6944

View on Pith Pith review arXiv 2014

[8] [8]

A. M. Odlyzko, Tables of zeros of the Riemann zeta function,https://www-users .cse.umn.edu/~odlyzko/zeta_tables/index.html

View on Pith

[9] [9]

S. Ramanujan, Highly composite numbers,Proc. London Math. Soc.(2)14(1915), 347–409. doi:10.1112/plms/s2_14.1.347

View on Pith doi:10.1112/plms/s2_14.1.347 1915

[10] [10]

S. Ramanujan, Highly composite numbers, annotated and with a foreword by J.- L. Nicolas and G. Robin,Ramanujan J.1(1997), 119–153; published as “Highly Composite Numbers by Srinivasa Ramanujan.” doi:10.1023/A:1009764017495

View on Pith doi:10.1023/a:1009764017495 1997

[11] [11]

G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Rie- mann,J. Math. Pures Appl.(9)63(1984), 187–213

View on Pith 1984

[12] [12]

X. Wu, Subsets of colossally abundant numbers, preprint (2019), arXiv:1903.03490. Dmitry R. Leshchiner email: leshch6@gmail.com Oleg R. Musin University of Texas Rio Grande Valley, Brownsville, TX 78520, USA. email: oleg.musin@utrgv.edu James Thompson email: james.thompson.5548@gmail.com 14

View on Pith Pith review arXiv 2019