Is this : $\lim \sup\frac{\sigma(n)}{n} , n \to\infty $ has a finite limit?

Question

The asymptotic growth rate of the sigma function can be expressed by :

$$\lim \sup\frac{\sigma(n)}{n\log(\log(n))}=e^{\gamma}$$ $$n \to\infty$$

according to the above limit , Is this : $$\lim \sup\frac{\sigma(n)}{n}$$ $$n \to\infty$$ has a finite limit ?

Note : I just would like to know the behavior of sigma function for large $n$ and to know the evaluation of second limit .

Thank you for any help .

Answer 1

the lim sup you are asking about is infinite

zeraoulia, the article you should read is Alaoglu and Erdos

http://www.renyi.hu/~p_erdos/1944-03.pdf

The numbers that are most efficient at giving large ( and increasing without bound) $\sigma(n)/n$ are called Colossally Abundant Numbers there. See https://mathoverflow.net/questions/79927/which-n-maximize-gn-frac-sigmann-log-log-n

I should also point out that programming the CA numbers is not so easy. However, the approximate factorization of one of these is very similar to a fairly easy sequence, $$ \operatorname{lcm} \; \{1,2,3,4,5, \ldots, N \} $$ Note that this number increases only when $N$ is a prime or prime power, so make the sequence under consideration from those.

Furthermore, you ask about the ratio $\sigma(n)/n$ by itself. This is, as pointed out, unbounded. As you can see at How would I find a number where $\sum_{d\mid n}d > 100n$? we can make arbitrarily large values from either the factorials or the primorials. I decided to do it with CA numbers, I did get the ratio up to $30$ and gave up. I think someone dedicated could do the task there with the LCM numbers I describe above.

Prime Power : 1  ratio:  1     n : 1      sigma(n) :  1
Prime Power : 2  ratio:  1.5     n : 2      sigma(n) :  3
Prime Power : 3  ratio:  2     n : 6      sigma(n) :  12
Prime Power : 4  ratio:  2.33333     n : 12      sigma(n) :  28
Prime Power : 5  ratio:  2.8     n : 60      sigma(n) :  168
Prime Power : 7  ratio:  3.2     n : 420      sigma(n) :  1344
Prime Power : 8  ratio:  3.42857     n : 840      sigma(n) :  2880
Prime Power : 9  ratio:  3.71429     n : 2520      sigma(n) :  9360
Prime Power : 11  ratio:  4.05195     n : 27720      sigma(n) :  112320
Prime Power : 13  ratio:  4.36364     n : 360360      sigma(n) :  1572480
Prime Power : 16  ratio:  4.50909     n : 720720      sigma(n) :  3249792
Prime Power : 17  ratio:  4.77433     n : 12252240      sigma(n) :  58496256
Prime Power : 19  ratio:  5.02561     n : 232792560      sigma(n) :  1169925120
Prime Power : 23  ratio:  5.24412     n : 5354228880      sigma(n) :  28078202880
Prime Power : 25  ratio:  5.41892     n : 26771144400      sigma(n) :  145070714880
Prime Power : 27  ratio:  5.55787     n : 80313433200      sigma(n) :  446371430400
Prime Power : 29  ratio:  5.74952     n : 2329089562800      sigma(n) :  13391142912000
Prime Power : 31  ratio:  5.93499     n : 72201776446800      sigma(n) :  428516573184000
Prime Power : 32  ratio:  6.03071     n : 144403552893600      sigma(n) :  870856261632000
Prime Power : 37  ratio:  6.1937     n : 5342931457063200      sigma(n) :  33092537942016000
Prime Power : 41  ratio:  6.34477     n : 219060189739591200      sigma(n) :  1389886593564672000
Prime Power : 43  ratio:  6.49232     n : 9419588158802421600      sigma(n) :  61155010116845568000
Prime Power : 47  ratio:  6.63046     n : 442720643463713815200      sigma(n) :  2935440485608587264000
Prime Power : 49  ratio:  6.74886     n : 3099044504245996706400      sigma(n) :  20915013459961184256000
Prime Power : 53  ratio:  6.8762     n : 164249358725037825439200      sigma(n) :  1129410726837903949824000
Prime Power : 59  ratio:  6.99274     n : 9690712164777231700912800      sigma(n) :  67764643610274236989440000
Prime Power : 61  ratio:  7.10738     n : 591133442051411133755680800      sigma(n) :  4201407903837002693345280000
Prime Power : 64  ratio:  7.16378     n : 1182266884102822267511361600      sigma(n) :  8469504822020624477061120000
Prime Power : 67  ratio:  7.27071     n : 79211881234889091923261227200      sigma(n) :  575926327897402464440156160000
Prime Power : 71  ratio:  7.37311     n : 5624043567677125526551547131200      sigma(n) :  41466695608612977439691243520000
Prime Power : 73  ratio:  7.47411     n : 410555180440430163438262940577600      sigma(n) :  3068535475037360330537152020480000
Prime Power : 79  ratio:  7.56872     n : 32433859254793982911622772305630400      sigma(n) :  245482838002988826442972161638400000
Prime Power : 81  ratio:  7.63179     n : 97301577764381948734868316916891200      sigma(n) :  742585584959041199989990788956160000
Prime Power : 83  ratio:  7.72374     n : 8076030954443701744994070304101969600      sigma(n) :  62377189136559460799159226272317440000
Prime Power : 89  ratio:  7.81053     n : 718766754945489455304472257065075294400      sigma(n) :  5613947022290351471924330364508569600000
Prime Power : 97  ratio:  7.89105     n : 69720375229712477164533808935312303556800      sigma(n) :  550166808184454444248584375721839820800000
Prime Power : 101  ratio:  7.96918     n : 7041757898200960193617914702466542659236800      sigma(n) :  56117014434814353313355606323627661721600000
Prime Power : 103  ratio:  8.04655     n : 725301063514698899942645214354053893901390400      sigma(n) :  5836169501220692744588983057657276819046400000
Prime Power : 107  ratio:  8.12175     n : 77607213796072782293863037935883766647448772800      sigma(n) :  630306306131834816415610170226985896457011200000
Prime Power : 109  ratio:  8.19626     n : 8459186303771933270031071135011330564571916235200      sigma(n) :  69333693674501829805717118724968448610271232000000

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Decided to do it myself this way with the LCM, and see how far I can get. Below is an abbreviated printout. I told it to just print when the ratio became at least as large as the next integer...

Prime Power : 1 prime : 1 floor of ratio:  1 log base ten of n :  0
Prime Power : 3 prime : 3 floor of ratio:  2 log base ten of n :  0.778151
Prime Power : 7 prime : 7 floor of ratio:  3 log base ten of n :  2.62325
Prime Power : 11 prime : 11 floor of ratio:  4 log base ten of n :  4.44279
Prime Power : 19 prime : 19 floor of ratio:  5 log base ten of n :  8.36697
Prime Power : 32 prime : 2 floor of ratio:  6 log base ten of n :  14.1596
Prime Power : 61 prime : 61 floor of ratio:  7 log base ten of n :  26.7717
Prime Power : 103 prime : 103 floor of ratio:  8 log base ten of n :  44.8605
Prime Power : 169 prime : 13 floor of ratio:  9 log base ten of n :  73.6195
Prime Power : 289 prime : 17 floor of ratio:  10 log base ten of n :  127.488
Prime Power : 509 prime : 509 floor of ratio:  11 log base ten of n :  223.273
Prime Power : 881 prime : 881 floor of ratio:  12 log base ten of n :  383.769
Prime Power : 1531 prime : 1531 floor of ratio:  13 log base ten of n :  665.019
Prime Power : 2671 prime : 2671 floor of ratio:  14 log base ten of n :  1153.3
Prime Power : 4639 prime : 4639 floor of ratio:  15 log base ten of n :  2010.98
Prime Power : 8089 prime : 8089 floor of ratio:  16 log base ten of n :  3507.89
Prime Power : 14107 prime : 14107 floor of ratio:  17 log base ten of n :  6130.6
Prime Power : 24649 prime : 157 floor of ratio:  18 log base ten of n :  10714

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Answer 2

No. For example, one has $\frac{\sigma(n!)}{n!} > H_n > \ln(n)$, with $H_n$ being a harmonic number.

To prove this, consider that for all $1 \leq k \leq n$ we have $k \mid n!$, so also $\frac{n}{k} \mid n!$.

Therefore the sequence $\frac{\sigma(k)}{k}$ contains a subsequence that goes to infinity, so the lim sup is infinity as well, therefore it is not finite. Note that we could see this from the original limit as well, but the above gives a nice elementary proof of it.

Answer 3

If $\sigma(n) < C n$ for a constant $C$ then $\limsup_{n \to \infty} \frac{\sigma(n)}{n \ln \ln n} = 0$. Contradiction.