Highly composite numbers andchampions
Jean-Louis Nicolas

1 Introduction
PLet d(n) = a|n 1 be the number of divisors of n. In his PHD thesis entitled Highly composite numbers (cf. [23, 18]), Srinivasa Ramanujan defines aninteger n to be hignly composite (hc for short)
if m < n implies d(m) < d(n).
More generally, if the sequence f (n) n>1 satisfies
lim sup f (n) = ∞ and for all ε > 0,n→∞

lim f (n)/nε = 0,n→∞(1)

one says that n is a f -champion if m < n implies f (m) < f (n) so that a hc
number is a d-champion.
Here some properties of hc numbers are presented first, following which
certain f-champions will be discussed.
Let us set p1 = 2, p2 = 3, . . . , pk the k-th prime. An integer n is said with
non increasing exponent (wnie for short) if vpk (n), the largest number suchthat pvpk (n) divides n, is a non-increasing function on pk . Ramanujan provedthat a hc number is a wnie number (cf. [23, Section 8]). He was consequently
interested by these wnie numbers and wrote with Hardy a nice article (cf.
[12]) on the number Qwnie (x) of wnie numbers up to x where they prove thatfor x tending to infinitys!
2πlog xQwnie (x) = exp (1 + o(1)) √.
(2)
3 log log x
Let us call Qf (x) the number of f -champions up to x. Therefore Qd (x) is thenumber of hc numbers 6 x. From the inequality Qd (x) 6 Qwnie (x) it follows
that (2) provides an upper bound for Qd (x). In [23, Section 28], Ramanujanproves that12

Jean-Louis NicolasQd (x) = o√(log x) log log x(log log log x)3/2

does not hold.
Let us define c(x) by Qd (x) = (log x)c(x) . It is now known that (cf. [18,Section V] and [10, 17, 19])
c1 = lim sup c(x) 6 1.71

andx→∞

c2 = lim inf c(x) ∈ [1.136, 1.44]
x→∞

(3)

and conjectured that limx→∞ c(x) = (log 30)/(log 16) = 1.226 . . . No effective
upper or lower bounds for Qd (x) of the form A logγ x is known.
Let n be a hc number. As n is wnie, it can be written
n=KYpakkwith

a1 > a2 > . . . > aK .

k=1

Ramanujan proved (cf. [23, Equation (43)]) that if n ∈/ {4, 36}, then apK =1 holds and, if k is fixed and n tends to infinity, then (cf. [23, Equation (86)] apk (log pk ) = (1 + o(1))(log pK /(log 2). He also proved thatlimn→∞ pK /(log n) = 1. Other inequalities about the factorization of a hc
number into primes are given in [23, Sections 8-27], while, in [23, Section 28],
it is proved that, as hc → ∞, the quotient of two consecutive hc numbers
tends to 1.
For f satisfying (1), an integer N is called a super f -champion if thereexists ε > 0 such thatf (N )f (M )
6MεNεfor M 6 N

andf (M )f (N )<MεNεfor M > N.

(4)

The super d-champions were introduced by Ramanujan who called them superior highly composite (shc) numbers (cf. [23, Section 32]), and this was avery great idea. The number ε is called a parameter of the super f -champion
N . Clearly, a super f -champion is an f -champion.
For ε > 0 fixed, the conditions (1) imply that the sequence f (n)/nε has amaximum which is attained at a finite number of points. If it is attained atonly one point, say N , from (4), N is a super f -champion. If it is attained atseveral points, the largest one is super f -champion.
For ε > 0, the numberYNε =pvp (N ) with vp (N ) = b1/(pε − 1)c(5)pis a super d-champion (i.e. is shc) of parameter ε (cf. [21, Equation (3.8)]).
Let us consider the setlog(1 + 1/k)
E=, k > 1, p prime
(6)
log p

Highly composite numbers and champions

3and order the elements of E ∪ {∞} in the decreasing sequence
ε1 = ∞ > ε2 = 1 > ε3 =log 2
log 3/2> ε4 => . . . > εi > . . .
log 3
log 2

(7)

In [21, Sections 3.1–3.2] (see also the last lines of Section 3 of [2]), it isexplained that there could exist elements in the set E defined by (6) admitting
two representations
εi =log(1 + 1/ki0 )log(1 + 1/ki )=log qi
log qi0(8)with ki > ki0 > 1 and qi < qi0 , but not three. An element εi ∈ E satisfying (8)is said to be extraordinary. If εi is not extraordinary, it is said to be ordinaryand satisfies in only one way
εi =log(1 + 1/ki ).
log qi(9)Let εi−1 and εi be two consecutive elements of the sequence (7) and ε anumber satisfting εi−1 > ε > εi . Then, from (5), Nε = Nεi−1 , and there is
only one shc number of parameter ε (cf. [21, Lemma 3.6]), namely
Nε = Nεi−1 .
For i > 1, the number Nεi is shc and its parameters are all the ε’s belonging
to the interval [εi , εi−1 ]. Moreover, if εi is ordinary and satisfies (9), then,from (5),Nεi = qi Nεi−1 ,while if εi is extraordinary and satisfies (8), then, from (5),Nεi = qi qi0 Nεi−1 ,which allows an easy computation of shc numbers. If εi is ordinary, then themaximum of d(n)/nεi is attained at two numbers Nεi−1 and Nεi . If εi is
extraordinary and satisfies (8), then the maximum of d(n)/nεi is attained atfour numbers Nεi−1 , qi Nεi−1 , qi0 Nεi−1 , Nεi .
All εi > (log 2)/ log 109 = 0.0334 . . . are ordinary. One may conjecture thatextraordinary numbers do not exist, but it seems very difficult to prove it.
Ramanujan defined the hc numbers in order to study how large the numberof divisors of an integer n can be. Let β = √log(3/2)/
P log 2= 0.584 . . ., li(t)the logarithmic integral of t and R(t) = 2 t + ρ tρ /ρ2 / log2 t, where ρruns over the non-trivial zeros of the Riemann ζ function. In [23, Equation(236], for n tending to infinity, it is proved

4Jean-Louis Nicolaslogβ nlog d(n)6 li(log n) + β li(logβ n) −− R(log n) + O
log 2log log n√
log n.
(log log n)3(10)Inequality (10) is an equality when n is shc. An effective form of (10) is given
in [21].
In the last section of [23], Ramanujan considers large values of the iterated function d(d(n)) and proves that there exist infinitely many intgers nsatisfying√√√log nlog nlog d(d(n)) > ( 2 log 4)= 1.96 . . .
.
log log n
log log n
In [6], this result has been improved :1/2∞(log d(d(n))) log log n  X 2√= 8log (1 + 1/j)
= 2.79 . . .
lim suplog nn→∞j=1
Since some fundamental properties of hc numbers (i.e. d-champions) havebeen discussed, now more generally, certain f-champions will be considered.

2 Champion numbers and the Riemann hypothesis
PLet σ(n) = δ|n δ be the sum of the divisors of n and γ = 0.577 . . . the Euler
constant. In 1982, Guy Robin proved that, under the Riemann hypothesis,forn > 5040,

σ(n)/n < eγ log log n(11)holds and, moreover, that (11) is equivalent to the Riemann hypothesis (cf.
[26, 27]). The main tool of his proof is the use of the super champion numbers
for the function σ(n)/n. These numbers were introduced in 1944 (cf. [2]) andcalled colossally abundant (CA for short) by Alaoglu and Erdős who did notknow that, earlier, in a manuscript not yet published, Ramanujan alreadydefined these numbers and called them generalized superior highly composite(cf. [24, Section 59]). The CA numbers have been used in many papers andmore especially in [16, 11, 27] and in the book [5]. The inequality (11) andits equivalence with the Riemann hypothesis have aroused many papers, cf.
[4, 7, 8, 9, 13, 14, 15, 28, 29].
In his PHD thesis, Srinivasa Ramanujan worked on the large values takenby the function σ(n). In the notes of the book Collected Papers of Srinivasa
Ramanujan, about the paper Highly Composite Numbers (cf. [23]), it is mentionned “ The London Mathematical Society was in some financial difficultyat the time, and Ramanujan suppressed part of what he had written in orderto save expense “. After the death of Ramanujan, all his manuscripts wereHighly composite numbers and champions

5sent to the University of Cambridge (England) where they slept in a closet
during a long time. They reappeared in the eighthy’s and, among them, handwritten by Ramanujan, (see [25, p. 280–312]) the suppressed part of “HighlyComposite Numbers”. A typed version can be found in [24] or in [3, Chap. 8].
For the history of this manuscript, see the foreword of [24], the introductionof Chapter 8 of [3] and [20].
In this suppressed part, under the Riemann hypothesis, Ramanujan gave
the asymptotic upper bound (cf. [24, Equation (382)])√2( 2 − 1)
O(1)
σ(n)
γ
6 e log log n − √+ S1 (log n) + √(12)
nlog nlog n log log nwith (cf. [24, Section 65]),
S1 (x) = −Xρxρ−11 X xρ=ρ(ρ − 1)x ρ |ρ|2

where ρ runs over the non-trivial zeros of the Riemann ζ function. In [22], aneffective form of (12) is given that yields a slight improvment of (11), namely,
forn > 5040,

σ(n)0.095 .
< eγ log log n − √nlog n

Let us set f (n) = σ(n)/n. The f -champion numbers were called superabundant by Alaoglu and Erdős in [2, Section 2] and earlier generalized highly
composite by Ramanujan (cf. [24, Section 59]). In [11], it is proved that thenumber Qf (x) of superabundant numbers up to x satisfies Qf (x) > log1+a x
for a < 5/48 and x large enough. It is not known
whether there exists apositive constant b such that Qf (x) = O logb x .
The σ-champions were called hignly abundant by Alaoglu and Erdős in[2, Section 5]. They give some properties of these numbers and ask questions
about the size of Qσ (x) or about the factorization into primes of σ- champion
numbers which, up to now, have not been answered.

3 Highly composite polynomials
Let Fq be the finite field with q elements. It is well known that the arithmeticof polynomials of Fq [t] looks like the arithmetic of integers. In [1], Ardavan
Afshar considers the question of maximising the divisor function in Fq [t] witha special attention to the case q = 2. Here, for simplicity, only the case q = 2
is considered, but the results for q > 2 are similar.
Let M = {f ∈ F2 [t]}, Mn = {f ∈ M, deg f = n},PI = {f ∈M irreducible}, Is = {f ∈ I, deg f = s} and π(s) = |Is | = 1s δ|s µ(δ)2s/δ ,6Jean-Louis Nicolas

where µ(δ) is the Möbius function. For f ∈ M, let τ (f ) be the number ofdivisors of f , and observe that a generic polynomial f in M is of the formYXYf=pap with deg f =ap deg p and τ (f ) =(1 + ap )
p∈I

p∈I

p∈I

where only finitely many ap ’s are non-zero.
If deg f = n, then f is said to be highly composite (hc) if, for all polynomials g of degree 6 n, τ (g) 6 τ (f ) holds. At the end of [1], a table of hc
polynomials of degrees 6 39 is given.
A polynomial h = h(x) ∈ M is called a x-superior highly composite, orjust x-shc, if for all f ∈ M(> τ (f )/q (deg f )/x if deg h > deg f
τ (h)(degh)/xq> τ (f )/q (deg f )/x if deg h < deg f.
The polynomialh(x) =Y Yk>1 p∈Ikakp

with ak =12k/x − 1

.

(13)

is x-shc. A polynomial h is called x-semi-superior highly composite or x-sshc,if for all f ∈ Mτ (h)τ (f )> degf /x .
degh/xqq
A polynomial that is x-shc or x-sshc for some x > 0 is called superior highlycomposite (shc) or semi-superior highly composite (sshc), respectively. It iseasy to see that a shc polynomial is sshc and that a sshc polynomial is hc.
Here, one defines the set
s log 2S=, s, r > 1 .
log(1 + 1/r)
Note that if x ∈ S, then there is a unique pair (s, r) such that x =(s log
 2)/ log(1 + 1/r) (cf. [1, Lemma 2.4]). The set S is ordered in a sequencexi i>1 :x1 = 1 < x2 =log 2
si log 2< x3 = 2 < . . . < xi =< ...
log(3/2)
log(1 + 1/ri )If xi = (si log 2)/ log(1 + 1/ri ) and x satisfies xi−1 6 x < xi then, from(13), theQ x-shc polynomial h(x) is equal to h(xi−1 ). Moreover, h(xi ) =h(xi−1 ) p∈Is p.
i

If x ∈/ S, then the maximum of τ (f )/2(degf )/x for f ∈ M is attained atonly one polynomial. If x = xi ∈ S, then this maximum is attained at 2π(s)Highly composite numbers and champions

7

polynomials, namelyh(xi )with 0 6 v 6 π(s), 1 6 j1 < . . . < jv and j1 , . . . , jv ∈ Is .
pj1 pj2 . . . pjv
The situation is different from the one of shc numbers, where the maximum ofd(n)/nε is attained at 1, 2 and possibly 4 (if extraordinary ε’s exist) numbers.
As π(s) tends to infinity with s, the maximum of τ (f )/2(deg f )/x can beattained at an unbounded number of polynomials.
It would be interesting to have an estimate of the number of hc polynomialsof degree 6 n.

References
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2. Alaoglu, L., Erdős, P.: On highly composite and similar numbers. Trans. Amer. Math.
Soc. 56 448–469 (1944)
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Notices (2012) 4051-4061
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8Jean-Louis Nicolas

18. Nicolas, J.-L.: On highly composite numbers. In: Andrews, G. E., Askey, R. A., Berndt,
B. C., Ramanathan, K.G., Rankin, R.A. (eds.) Ramanujan Revisited, Proceedings ofthe Centenary Conference, University of Illinois at Urbana-Champaign 215–244 (1987). Academic Press, New-York-London (1988)
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University Press (1927)
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number theory, Paris (1981–82) 233–244. Progr. Math., 38, Birkhäuser, Boston, (1983)
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Jean-Louis Nicolas,Univ. Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208
Institut Camille Jordan, Mathématiques, Bât. Doyen Jean Braconnier,43 Bd du 11 Novembre 1918, F-69622 Villeurbanne cedex, France.
nicolas@math.univ-lyon1.fr,
http://math.univ-lyon1.fr/homes-www/nicolas/