Contemporary Mathematics
Volume 627, 2014
http://dx.doi.org/10.1090/conm/627/12539

Ramanujan, Robin, highly composite numbers, andthe Riemann Hypothesis
Jean-Louis Nicolas and Jonathan Sondow
Abstract. We provide an historical account of equivalent conditions for theRiemann Hypothesis arising from the work of Ramanujan and, later, Guy
Robin on generalized highly composite numbers. The ﬁrst part of the paper
is on the mathematical background of our subject. The second part is on its
history, which includes several surprises.

1. Mathematical BackgroundDeﬁnition. The sum-of-divisors function σ is deﬁned by1
σ(n) :=.
d=ndd|nd|nIn 1913, Grönwall found the maximal order of σ.
Grönwall’s Theorem [8]. The functionσ(n)G(n) :=(n > 1)
n log log nsatisﬁeslim sup G(n) = eγ = 1.78107 . . . ,n→∞where
11γ := lim 1 + + · · · + − log n = 0.57721 . . .
n→∞2n
is the Euler-Mascheroni constant.
Grönwall’s proof uses:
Mertens’s Theorem [10]. If p denotes a prime, then−11 1lim= eγ .
1−x→∞ log xpp≤x

2010 Mathematics Subject Classiﬁcation. Primary 01A60, 11M26, 11A25.
Key words and phrases. Riemann Hypothesis, Ramanujan’s Theorem, Robin’s Theorem,
sum-of-divisors function, highly composite number, superabundant, colossally abundant, Eulerphi function.
©2014 American Mathematical Society

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JEAN-LOUIS NICOLAS AND JONATHAN SONDOW

Figure 1. Thomas Hakon GRÖNWALL (1877–1932)

Figure 2. Franz MERTENS (1840–1927)
Now we come to:Ramanujan’s Theorem [2, 15, 16]. If RH is true, then for n0 large enough,n > n0 =⇒ G(n) < eγ .
To prove his theorem, Ramanujan introduces a real non-negative parameter s,
considers the multiplicative function n → σ−s (n) = d|n d−s , and calls an integer
N a generalized highly composite number ifN  < N =⇒ σ−s (N  ) < σ−s (N ).
When s = 1 these numbers have been called superabundant by Erdős and Alaoglu[1], while for s = 1 they have only been studied by Ramanujan. Further, Ramanujan

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RAMANUJAN, ROBIN, AND THE RIEMANN HYPOTHESIS

147

Figure 3. Srinivasa RAMANUJAN (1887–1920)
calls an integer N a generalized superior highly composite number of parameter ε > 0if
σ−s (N )σ−s (N  )≥N  < N =⇒εN(N  )εandσ−s (N )σ−s (N  )N  > N =⇒>.
εN(N  )εWhen s = 1 these numbers have been called colossally abundant by Erdős andAlaoglu. It is easily seen that all generalized superior highly composite numbers
are generalized highly composite.
The prime factorization of a generalized superior highly composite number N
can be obtained from the value of the parameter ε. For r = 1, 2, 3 . . ., Ramanujan
deﬁnes xr by−s(r+1)
1 − xrxεr =1 − x−srr
and thenReϑ(xr )N=r=1

where ϑ(x) = p≤x log p denotes Chebyshev’s function and R is the largest integer
such that xR ≥ 2. One has

σ−s (N ) =R 1 − p−s(r+1)
r=1 p≤xr

1 − p−sr,and, to estimate σ−s (N ), one needs an estimate of  ∞ log p1dtlog 1 − s = −ppt − 1p≤xp≤x s
pwhence the idea of considering the sum p≤x plogs −1 ·
Here is an excerpt from Ramanujan’s proof: Ramanujan [16, p. 133]: . . . assume that . . . s > 0 . . . if p is the largest prime not greater than x, thenlog 2log 3log 5log p+ s+ s+ ··· + s
2s − 13 −1
5 −1p −1 θ(x)dxx − ϑ(x)dx + O{x−s (log x)4 }.
−s=C+sx −1x1−s (xs − 1)2

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JEAN-LOUIS NICOLAS AND JONATHAN SONDOW

But it is known thatx − θ(x) =√1x + x3 + xρ
ρ− x 12 ρ
ρ

1+ O(x 5 )where ρ is a complex root of ζ(s). . . .
The last equation is a variant of the classical explicit formula in prime number
theory. This shows “explicitly” how Ramanujan used RH in his proof.
From the estimate for σ−s (N ), Ramanujan deduces that, for 1/2 < s < 1 andall suﬃciently large integers n, the upper bound
112s(2 2s − 1) (log n) 2 −sσ−s (n) ≤ |ζ(s)| exp Li((log n)1−s ) −2s − 1
log log n1
 (log n)ρ−ss(log n) 2 −s−+O
log log n ρ ρ(ρ − s)(log log n)2holds. Finally, making s tend to 1, he gets√
lim sup (σ−1 (n) − eγ log log n) log n ≤ −eγ (2 2 − 4 − γ + log 4π) = −1.393 . . . < 0.
n→∞

Since G(n) = σ−1 (n)/ log log n, this proves Ramanujan’s Theorem.
Next we have:Robin’s Theorem [18, 19]. RH is true if and only if n > 5040 (= 7!) =⇒G(n) < eγ .

Figure 4. Guy ROBIN (photo courtesy of Guy Robin)This is a free offprint provided to the author by the publisher. Copyright restrictions may apply.

RAMANUJAN, ROBIN, AND THE RIEMANN HYPOTHESIS

149

Figure 5. Robin’s paper on σ and RH, Journal de Mathématiques
Pures et Appliquées, 1984

To prove his theorem, Robin uses generalized superior highly composite numbers only with s = 1, i.e., colossally abundant numbers (CA for short). First, heshows that if N  < N  are two consecutive CA numbers, thenN  < n < N  =⇒ G(n) ≤ max(G(N  ), G(N  )).
Second, by numerical computation, he checks that G(N ) < eγ for all integers Nwith 5041 < N < 55440, as well as for all CA numbers N ≥ 55440 whose largest
prime factor P + (N ) is < 20000.
Further, if a CA number N satisﬁes P + (N ) > 20000, then getting an upperbound for σ−1 (N ) requires a precise estimate of the Mertens product−11.
1−pp≤xThe sum-of-divisors function σ and Euler’s totient function φ, deﬁned as
1φ(n) :=
1=n1−,
p1≤k≤np|n(k,n)=1

are related by the inequalities
σ(n) φ(n)6·< 1,<π2nnThis is a free offprint provided to the author by the publisher. Copyright restrictions may apply.

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JEAN-LOUIS NICOLAS AND JONATHAN SONDOW

which hold for all n > 1. Mertens’s Theorem implies that the minimal order of φis given byn/ log log n= eγ .
lim supφ(n)n→∞To estimate the Mertens product, Robin used ideas from a result on the φ function proved by his thesis advisor Nicolas in 1983.
Nicolas’s Theorem [11, 12]. RH is true if and only ifprime p > 2 =⇒p#/ log log p#> eγ ,φ(p#)

where p# := 2 · 3 · 5 · 7 · 11 · · · p denotes a primorial.

Figure 6. Jean-Louis NICOLAS (photo courtesy of Jean-Louis Nicolas)
Nicolas in turn used Landau’s Oscillation Theorem [9], which Landau hadapplied√ in 1905 to prove Chebyshev’s bias in the form π(x; 4, 3) − π(x; 4, 1) =Ω+ ( x/ log x).

Figure 7. Edmund Georg Hermann LANDAU (1877–1938)

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RAMANUJAN, ROBIN, AND THE RIEMANN HYPOTHESIS

151

Recently, Caveney and the authors gave two reformulations of Robin’s Theorem.
Caveney, Nicolas, and Sondow’s Theorems [5, 6].
Deﬁne an integer N > 1 to be a GA1 number if N is composite and G(N ) ≥G(N/p) for all prime factors p. Call an integer N a GA2 number if G(N ) ≥ G(aN )for all multiples aN . Then:
1. RH is true if and only if 4 is the only number that is both GA1 and GA2.
2. A GA2 number N > 5040 exists if and only if RH is false, in which case N iseven and > 10 8576 .

Figure 8. Geoﬀrey CAVENEY (photo courtesy of Geoﬀrey Caveney)

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152JEAN-LOUIS NICOLAS AND JONATHAN SONDOW

2. Historical Survey
Our story begins in 1915, when Ramanujan published the ﬁrst part of his dissertation “Highly Composite Numbers” (HCN for short).

Figure 9. Ramanujan’s HCN Part 1, Proceedings of the London
Mathematical Society, 1915
Ramanujan (in [14]): I deﬁne a highly composite number as a number whose number of divisors exceeds that of all its predecessors.
His thesis was written at Trinity College, University of Cambridge. Accordingto The Mathematics Genealogy Project [13], his advisors were Hardy and Littlewood. However, Littlewood served in World War I and was not in Cambridge foralmost all of the time that Ramanujan was there.
In 1944, Erdős published a paper “On highly composite and similar numbers”
with Alaoglu [1].
Erdős (in “Ramanujan and I” [7]): Ramanujan had a very long manuscript onhighly composite numbers but some of it was not published due to a paper shortage
during the First World War.
Dyson (email to Sondow, 2012): Hardy told me, “Even Ramanujan could not makehighly composite numbers interesting.” He said it to discourage me from workingon H. C. numbers myself. I think he was right.

Figure 10. Freeman John DYSON (photo courtesy of Freeman Dyson)
In 1982 Rankin published a paper on “Ramanujan’s manuscripts and notebooks.” He quoted Hardy’s mention of “the suppressed part of HCN” in a 1930

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RAMANUJAN, ROBIN, AND THE RIEMANN HYPOTHESIS

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letter to Watson.
Rankin (in [17]): The most substantial manuscript consists of approximately 30
pages of HCN carrying on from where the published paper stops.
By a curious coincidence, 1981–1982 is also the year of Séminaire Delange-PisotPoitou’s exposition [18] of Robin’s Theorem, in which he improved on Ramanujan’s
Theorem without ever having heard of it!
Berndt (email to Sondow, 2012): It is doubtful that Rankin took notice of Robin’s
paper. I deﬁnitely did not.
After I began to edit Ramanujan’s notebooks, I wrote Trinity College in 1978
for a copy of the notes that Watson and Wilson made in their eﬀorts to edit thenotebooks. I also decided to write for copies of all the Ramanujan material that wasin the Trinity College Library. Included in their shipment to me was the completionof Ramanujan’s paper on highly composite numbers. I put all of this on display
during the Ramanujan centenary meeting at Illinois in June, 1987.

Figure 11. Bruce Carl BERNDT holding Ramanujan’s slate(permission by Bruce Berndt)

Nicolas (email to Sondow, 2012): I keep a very strong souvenir of the conference organised in Urbana-Champaign in 1987 for the one hundred anniversary of Ramanujan. It is there that I discovered the hidden part of “Highly Composite Numbers”[ﬁrst published in [15], later in [16], and again in [2]].
What I have not written is that there was an error of calculus in Ramanujan’s
manuscript which prevented him from seeing Robin’s Theorem. Soon after discovering the hidden part, I read it and saw the diﬀerence between Ramanujan’s result
and Robin’s one. Of course, I would have bet that the error was in Robin’s paper,
but after recalculating it several times and asking Robin to check, it turned out that
there was an error of sign in what Ramanujan had written.
Thus it happened that Robin avoided the fate of the many mathematicians who
have found that (Berndt in [3, 4], quoting Gosper): Ramanujan reaches his handfrom his grave to snatch your theorems from you.

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JEAN-LOUIS NICOLAS AND JONATHAN SONDOW

Ramanujan’s Theorem was not explicitly stated by him in HCN Parts 1 or 2.
Nicolas and Robin formulated it for him in Note 71 of their annotated and correctedversion of HCN Part 2.
Nicolas and Robin (in [16]): It follows from (382) [(the corrected version of Ramanujan’s formula)] that under the Riemann hypothesis, and for n0 large enough,n > n0 =⇒ σ(n)/n < eγ log log n.
It has been shown in [19] that the above relation with n0 = 5040 is equivalent to theRiemann hypothesis.
Here [19] is Robin’s paper, which he published three years before learning ofRamanujan’s Theorem. However, a reader of [16] who neglects to look up [19] in theReferences is left with the misimpression “that the above relation with n0 = 5040 isequivalent to the Riemann hypothesis” was proven after whoever proved it learnedof “the above relation”!
In 1993, HCN Part 2 was submitted to Proceedings of the London MathematicalSociety, which had published Part 1 in 1915. The paper was accepted, but could
not be published, because Trinity College did not own the rights to Ramanujan’s
papers and was not able to obtain permission from his widow, Janaki.
Janaki passed away in 1994, and HCN Part 2 was eventually published byAlladi in the ﬁrst volume of his newly-founded Ramanujan Journal. Later, thepaper was republished as Chapter 9 in the book by Andrews and Berndt [2], who
expanded the commentary and brought it up-to-date.

Figure 12. Krishnaswami ALLADI, founder of The Ramanujan
Journal (photo courtesy of Krishnaswami Alladi)

Here our story ends. If it has oﬀended anyone, we apologize.

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RAMANUJAN, ROBIN, AND THE RIEMANN HYPOTHESIS

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Figure 13. Ramanujan’s HCN Part 2, annotated by Nicolas andRobin, The Ramanujan Journal, 1997Figure 14. Jonathan SONDOW (left) and Ramjee RAGHAVAN,
Ramanujan’s grandnephew, by chance (!) seatmates on a ﬂight
to Charlotte, where S. took a connecting ﬂight to RAMA125 inGainesville and R. took one to a business meeting in Chicago(photo courtesy of Hisayo Foster)Acknowledgments
The authors are grateful to the anonymous referee for several valuable suggestions, and to Marc Deléglise for his help with typesetting the manuscript.

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JEAN-LOUIS NICOLAS AND JONATHAN SONDOW[2] George E. Andrews and Bruce C. Berndt, Ramanujan’s lost notebook. Part III, Springer,
New York, 2012. MR2952081[3] Bruce C. Berndt, Ramanujan reaches his hand from his grave to snatch your theorems fromyou, Asia Pac. Math. Newsl. 1 (2011), no. 2, 8–13. MR2817235[4] Bruce C. Berndt, Heng Huat Chan, Song Heng Chan, and Wen-Chin Liaw, Cranks—really,
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MR774171 (86f:11069)
Université de Lyon; CNRS; Université Lyon 1; Institut Camille Jordan, Mathématiques, 21 Avenue Claude Bernard, F-69622 Villeurbanne cedex, France
E-mail address: nicolas@math.univ-lyon1.fr
209 West 97th Street #6F, New York, New York 10025
E-mail address: jsondow@alumni.princeton.edu

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