Growth rate of sum of divisors cubed

Question

I a trying to find a result similar to:

$$\limsup_{n \to \infty} \frac{\sigma_1(n)}{n \log \log (n)} = e^\gamma$$

(where $\sigma_1$ is the sum of divisors function) but regarding the growth rate of $\sigma_3$ (apart from the trivial $\liminf$). I was not able to find any good reference around.

Thank you in advance.

Answer 1

(I found an answer!) As shown in T.H. Gronwall. Trans. Amer. Math. Soc. 14 (1913), 113–122,

$$\limsup_{n\to\infty} \frac{\sigma^k(n)}{n^k} = \zeta(k)$$

For $k>1$

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Question closed... here is Ramanujan's method for producing numbers for which $\sigma_3$ is surprisingly large. SEE:

https://en.wikipedia.org/wiki/Superior_highly_composite_number

https://oeis.org/A002201

https://simple.wikipedia.org/wiki/Colossally_abundant_number

https://oeis.org/A004490

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jagy@gost:~/Desktop/Cplusplus$ ./Colossally 3 0.000001  1234 


=========================================================  //  producing the deltas
 ( 0.1699250014423129 , 2 , 1 ) 
 ( 0.01989955743770461 , 2 , 2 ) 
 ( 0.002468255590751251 , 2 , 3 ) 
 ( 0.0003082352411876714 , 2 , 4 ) 
 ( 3.852477511632481e-05 , 2 , 5 ) 
 ( 4.815524557131172e-06 , 2 , 6 ) 
 ( 0.03310325630433738 , 3 , 1 ) 
 ( 0.00120322453294025 , 3 , 2 ) 
 ( 4.453334116799652e-05 , 3 , 3 ) 
 ( 1.649341164911445e-06 , 3 , 4 ) 
 ( 0.004950902167530941 , 5 , 1 ) 
 ( 3.944858479242779e-05 , 5 , 2 ) 
 ( 0.001496066099903359 , 7 , 1 ) 
 ( 4.355347520233727e-06 , 7 , 2 ) 
 ( 0.000313204965096843 , 11 , 1 ) 
 ( 0.0001774158143247915 , 13 , 1 ) 
 ( 7.183395239255508e-05 , 17 , 1 ) 
 ( 4.95113743709696e-05 , 19 , 1 ) 
 ( 2.621154625712154e-05 , 23 , 1 ) 
 ( 1.217631375969359e-05 , 29 , 1 ) 
 ( 9.774824236564204e-06 , 31 , 1 ) 
 ( 5.467300253590937e-06 , 37 , 1 ) 
 ( 3.907090067810915e-06 , 41 , 1 ) 
 ( 3.343993782981531e-06 , 43 , 1 ) 
 ( 2.501652345023562e-06 , 47 , 1 ) 
 ( 1.691797945951663e-06 , 53 , 1 ) 
 ( 1.194111689731658e-06 , 59 , 1 ) 
 ( 1.071705320372341e-06 , 61 , 1 ) 
=========================================================  // deltas in order and resulting numbers
 ( 0.1699250014423129 , 2 , 1 )    2 =  2
 ( 0.03310325630433738 , 3 , 1 )    6 =  2 3
 ( 0.01989955743770461 , 2 , 2 )    12 =  2^2 3
 ( 0.004950902167530941 , 5 , 1 )    60 =  2^2 3 5
 ( 0.002468255590751251 , 2 , 3 )    120 =  2^3 3 5
 ( 0.001496066099903359 , 7 , 1 )    840 =  2^3 3 5 7
 ( 0.00120322453294025 , 3 , 2 )    2520 =  2^3 3^2 5 7
 ( 0.000313204965096843 , 11 , 1 )    27720 =  2^3 3^2 5 7 11
 ( 0.0003082352411876714 , 2 , 4 )    55440 =  2^4 3^2 5 7 11
 ( 0.0001774158143247915 , 13 , 1 )    720720 =  2^4 3^2 5 7 11 13
 ( 7.183395239255508e-05 , 17 , 1 )    12252240 =  2^4 3^2 5 7 11 13 17
 ( 4.95113743709696e-05 , 19 , 1 )    232792560 =  2^4 3^2 5 7 11 13 17 19
 ( 4.453334116799652e-05 , 3 , 3 )    698377680 =  2^4 3^3 5 7 11 13 17 19
 ( 3.944858479242779e-05 , 5 , 2 )    3491888400 =  2^4 3^3 5^2 7 11 13 17 19
 ( 3.852477511632481e-05 , 2 , 5 )    6983776800 =  2^5 3^3 5^2 7 11 13 17 19
 ( 2.621154625712154e-05 , 23 , 1 )    160626866400 =  2^5 3^3 5^2 7 11 13 17 19 23
 ( 1.217631375969359e-05 , 29 , 1 )    4658179125600 =  2^5 3^3 5^2 7 11 13 17 19 23 29
 ( 9.774824236564204e-06 , 31 , 1 )    144403552893600 =  2^5 3^3 5^2 7 11 13 17 19 23 29 31
 ( 5.467300253590937e-06 , 37 , 1 )    5342931457063200 =  2^5 3^3 5^2 7 11 13 17 19 23 29 31 37
 ( 4.815524557131172e-06 , 2 , 6 )    10685862914126400 =  2^6 3^3 5^2 7 11 13 17 19 23 29 31 37
 ( 4.355347520233727e-06 , 7 , 2 )    74801040398884800 =  2^6 3^3 5^2 7^2 11 13 17 19 23 29 31 37
 ( 3.907090067810915e-06 , 41 , 1 )    3066842656354276800 =  2^6 3^3 5^2 7^2 11 13 17 19 23 29 31 37 41
 ( 3.343993782981531e-06 , 43 , 1 )    131874234223233902400 =  2^6 3^3 5^2 7^2 11 13 17 19 23 29 31 37 41 43
 ( 2.501652345023562e-06 , 47 , 1 )    6198089008491993412800 =  2^6 3^3 5^2 7^2 11 13 17 19 23 29 31 37 41 43 47
 ( 1.691797945951663e-06 , 53 , 1 )    328498717450075650878400 =  2^6 3^3 5^2 7^2 11 13 17 19 23 29 31 37 41 43 47 53
 ( 1.649341164911445e-06 , 3 , 4 )    985496152350226952635200 =  2^6 3^4 5^2 7^2 11 13 17 19 23 29 31 37 41 43 47 53
 ( 1.194111689731658e-06 , 59 , 1 )    58144272988663390205476800 =  2^6 3^4 5^2 7^2 11 13 17 19 23 29 31 37 41 43 47 53 59
 ( 1.071705320372341e-06 , 61 , 1 )    3546800652308466802534084800 =  2^6 3^4 5^2 7^2 11 13 17 19 23 29 31 37 41 43 47 53 59 61
=========================================================

2, 6, 12, 60, 120, 840, 2520, 27720, 55440, 720720, 12252240, 232792560, 698377680, 3491888400, 6983776800, 160626866400, 4658179125600, 144403552893600, 5342931457063200, 10685862914126400, 74801040398884800, 3066842656354276800, 131874234223233902400, 6198089008491993412800, 328498717450075650878400, 985496152350226952635200, 58144272988663390205476800, 3546800652308466802534084800, 


   sigma_index 3  delta bound  1e-06   prime bound   1234

Thu 01 Feb 2024 09:06:31 AM PST

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