Re: numbers n with large sum of divisors compared to n

David Bernier wrote:
The number-theoretical function sigma(n) is defined as
sigma(n) := sum_{ d | n} d .

For example, sigma(6) = 1+2+3+6 = 12.

The ratio sigma(n)/n can grow with n, but not very
fast if the Riemann Hypothesis is true.

According to Wikipedia,
http://en.wikipedia.org/wiki/Riemann_hypothesis#Consequences_of_the_Riemann_hypothesis


In 1984, Guy Robin published a paper:
"Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann"

Cf.:
http://en.wikipedia.org/wiki/Robin%27s_theorem#CITEREFRobin1984
(also has Lagarias' criterion).

Robin showed that the statement
"sigma(n) < exp(gamma) n log(log(n)) for all n >= 5041"
is equivalent to RH.
[ Ramanujan had proved RH ==>
sigma(n) < exp(gamma) n log(log(n)) for all sufficiently large n ].


To get an idea of what kinds of numbers have large sigma(n) relative
to n, I wrote a computer program to show the integers n >=5041
for which sigma(n)/( n log(log(n)) ) > 1.71 .
We have exp(gamma) = exp(0.577...) ~= 1.7810724179901979 .

This is what I have so far:
==============================

n = 7560, sigma(n) = 28800, ratio = 1.7399165192
n = 10080, sigma(n) = 39312, ratio = 1.7558143389
n = 15120, sigma(n) = 59520, ratio = 1.7385586743
n = 20160, sigma(n) = 79248, ratio = 1.7138106151
n = 25200, sigma(n) = 99944, ratio = 1.7124820364
n = 27720, sigma(n) = 112320, ratio = 1.7425367238
n = 30240, sigma(n) = 120960, ratio = 1.7139536874
n = 55440, sigma(n) = 232128, ratio = 1.7512465149
n = 65520, sigma(n) = 270816, ratio = 1.7178890011
n = 83160, sigma(n) = 345600, ratio = 1.7121096531
n = 110880, sigma(n) = 471744, ratio = 1.7348490103
n = 166320, sigma(n) = 714240, ratio = 1.7269287425
n = 277200, sigma(n) = 1199328, ratio = 1.7112437558
n = 332640, sigma(n) = 1451520, ratio = 1.7160969754
n = 360360, sigma(n) = 1572480, ratio = 1.7118721101
n = 720720, sigma(n) = 3249792, ratio = 1.7330653562
n = 1441440, sigma(n) = 6604416, ratio = 1.7277402116
n = 2162160, sigma(n) = 9999360, ratio = 1.7255702285
n = 2882880, sigma(n) = 13313664, ratio = 1.7106690213
n = 3603600, sigma(n) = 16790592, ratio = 1.7164672141
n = 4324320, sigma(n) = 20321280, ratio = 1.7235466599
n = 7207200, sigma(n) = 34122816, ratio = 1.7157791493
n = 10810800, sigma(n) = 51663360, ratio = 1.7160732878
n = 21621600, sigma(n) = 104993280, ratio = 1.7178999371
n = 24504480, sigma(n) = 118879488, ratio = 1.7117986886
n = 36756720, sigma(n) = 179988480, ratio = 1.7135828536
==============================

where ratio(n) := sigma(n)/( n log(log(n)) ) .


24504480 = 2^5 * 3^2 * 5 * 7 * 11 * 13 * 17 .

Looking at n = 360360, 720720 and 1441440 with largest
ratio of 1.73306 for the number 720720 suggests
there's an optimal allocation of '2' factors ...


I believe sigma is a multiplicative function.

With n = 288807105787200 ( = 64*27*25*7*11*13*17*19*23*29*31 )

I get sigma(n) = 1755535638528000 ( = 127*40*31*8*12*14*18*20*24*30*32 )

So Log( sigma(n)/n) =

log(127/64) + log(40/27) + log(31/25) + log(8/7) + log(12/11)
+ log(14/13) + log(18/17) + log(20/19) + log(24/23) + log(30/29)
+ log(32/31)

~ = 1.8047702852299

so sigma(n)/(n log(log(n)) ) ~= 1.734030266168 ,
and exp(gamma) ~= 1.781 .

David Bernier


Grönwall's theorem says that
lim sup_{ n -> oo} sigma(n)/( n log(log(n)) ) = exp(gamma).

Robin showed Robin's inequality ==> RH.

I wonder if it's known what kind of anomaly in the distribution
of the primes would allow a violation of Robin's inequality
(apart from what was clear from RH and what it says about
pi(x) - li(x) ) ...

David Bernier

.

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