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

In article <20090601.154257@xxxxxxxx>,
Rob Johnson <rob@xxxxxxxxxxxxxx> wrote:
In article <h01bj60tkh@xxxxxxxxxxxxxxxxx>,
David Bernier <david250@xxxxxxxxxxxx> wrote:
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 hypoth?se 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


Gr?nwall'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) ) ...

sigma(n) is multiplicative, so therefore is sigma(n)/n. Furthermore,
for any prime p,

sigma(p^k) p^{k+1}-1
---------- = --------- [1]
p^k p^k(p-1)

Thus,

sigma(p) p+1
-------- = --- [2]
p p

and

sigma(p^k) p
lim ---------- = --- [3]
k->oo p^k p-1

In any case, for k > 0,

p+1 sigma(p^k) p
--- <= ---------- < --- [4]
p p^k p-1

Therefore, knowing only the prime factors, but not their exponents
(except that each exponent is at least 1), we can estimate

sigma(n) --- p+1
-------- ~ | | sqrt( --- ) [5]
n p|n p-1

and this estimate is accurate to within a factor of

--- p^2
| | sqrt( ----- ) [6]
p|n p^2-1

Since the product over all primes

--- p^2
| | ----- = pi^2/6 [7]
p p^2-1

the estimate [5] is accurate to within a factor of pi/sqrt(6).

This means that by adjusting the exponents of the prime factors of n,
we can only affect the value of sigma(n)/n by at most a factor of
pi^2/6.

However, there can still be something gained by adjusting the
exponents since log(log(n)) also varies very slowly with n.

---
log(n) = > log(p^k) [8]
---

The change in log(n) is thus

---
d log(n) = > log(p) dk [9]
---

Using [1] and the multiplicativity of sigma(n)/n, we also have

sigma(n) --- p-p^{-k}
log -------- = > log -------- [10]
n --- p-1

The change in this is

sigma(n) --- log(p)
d log -------- = > ----------- dk [11]
n --- p^{k+1} - 1

The standard way to find, for a given set of primes, the exponents
that maximize [10] whlle holding [8] constant, is to have the vector
of coefficients of dk in [11] be a multiple of the vector of
coefficients of dk in [9]. In this case, that means setting k so
that 1/(p^{k+1} - 1) is a constant p^{k+1} is a constant for all
primes in our given set. Of course, this cannot be done exactly when
the exponents are integers, but it gives a target.

For example, let's try a factor of 4 and compute the exponents as
k = 4/log(p) - 1 (rounded to the nearest integer):

p k
2 5
3 3
5 1
7 1
11 1
13 1

n = 4324320
sigma(n) = 20321280
sigma(n)/(n log(log(n))) = 1.72354665986337

This is one of the numbers in the table above.

I was able to use this method with k = 23/log(p)-1 to yield a 1984902
digit number whose sigma was 1984904 digits and whose ratio was
1.7808890605380562. Close to e^gamma = 1.7810724179901980, yet still
below.

Rob Johnson <rob@xxxxxxxxxxxxxx>
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