2
$\begingroup$

Let $\mathbb{N}$ be the set of positive integers.

For $x \in \mathbb{N}$, $\sigma_1(x)$ gives the sum of the divisors of $x$. (For example, $\sigma_1(3) = 1 + 3 = 4$.)

We call the ratio $I(x) = \sigma_1(x)/x$ as the abundancy index of $x$.

Since $1 \leq x \mid x^2$, by a property of the abundancy index, we know that $I(x) \leq I(x^2)$, where equality holds if and only if $x = 1$.

My question is this: Is it possible to improve on the (equivalent) inequality $x\sigma_1(x) \leq \sigma_1(x^2)$ for $x \in \mathbb{N}$?

I am guessing it would suffice to consider composite $x$.

Update: Per Will Jagy's answer (and a subsequent comment by Erick Wong), we have the conjectured (sharp?) bounds

$$1 \leq \frac{I(x^2)}{I(x)} \leq \prod_{p}{\frac{p^2 + p + 1}{p^2 + p}} = \frac{\zeta(2)}{\zeta(3)} \approx 1.3684327776\ldots$$

I would still be interested in an (improved) lower bound for $I(x^2)/I(x)$ when $\omega(x) \geq 3$, where $\omega(x)$ is the number of distinct prime factors of $x$.

$\endgroup$
  • $\begingroup$ Related MSE question $\endgroup$ – Jose Arnaldo Bebita-Dris Jan 9 '15 at 16:27
  • 1
    $\begingroup$ It makes no difference how large you pick $\omega(x)$, if you just pick large enough prime factors you can shift the ratio $I(x^2)/I(x)$ down to as close to $1$ as you like. $\endgroup$ – Erick Wong Jan 11 '15 at 10:39
  • $\begingroup$ Okay, thanks @ErickWong! $\endgroup$ – Jose Arnaldo Bebita-Dris Jan 12 '15 at 7:50
2
$\begingroup$

The lower bound for $$ \frac{\sigma(x^2)}{x \sigma(x)} $$ is just $1,$ we can get arbitrarily close with $x=p$ for large prime $p$ or $x=pq$ with distinct large primes.

What remains is an upper bound for $$ \frac{\sigma(x^2)}{x \sigma(x)}, $$ which is number-theoretic "multiplicative" and is thereby subject to a method of Ramanujan, which takes time to work out. I think, already, having done dozens of analogous problems, that the optimum values occur at the primorials, those being the products of the consecutive primes from $2$ to some $p.$

The constant in the right hand column converges to something not very large; the primorial indicated is the one with largest prime factor in the second column, so 2, 6, 30, 210, etc.

        1      2   1.166666666666667
        2      3   1.263888888888889
        3      5   1.306018518518519
        4      7   1.329340277777778
        5     11   1.339411037457912
        6     13   1.346770438762626
        7     17   1.351171649346818
        8     19    1.35472736421352
        9     23   1.357181580453037
       10     29   1.358741559281144
       11     31   1.360111258433645
       12     37   1.361078620637368
       13     41   1.361869026340409
       14     43   1.362588830265536
       15     47   1.363192814676115
       16     53   1.363669122438056
       17     59   1.364054339704282
       18     61   1.364415009809968
       19     67   1.364714486326607
       20     71   1.364981449254292
       21     73   1.365234130011281
       22     79   1.365450148069827
       23     83   1.365645995767485
       24     89   1.365816488401164
       25     97   1.365960167812946
       26    101   1.366092759558997
       27    103   1.366220288756118
       28    107   1.366338514810873
       29    109    1.36645247131753
       30    113   1.366558545876707
       31    127   1.366642610747885
       32    131   1.366721643977746
       33    137    1.36679393434292
       34    139   1.366864170413544
       35    149   1.366925327647119
       36    151   1.366984883473524
       37    157   1.367039990499384
       38    163   1.367091129209189
       39    167   1.367139856437917
       40    173   1.367185273348968
       41    179   1.367227706163162
       42    181   1.367269210252354
       43    191   1.367306493989545
       44    193   1.367343011977205
       45    197   1.367378066661407
       46    199   1.367412422894238
       47    211   1.367442991892335
       48    223   1.367470367032307
       49    227   1.367496788517102
       50    229   1.367522752002739
       51    233   1.367547834038468
       52    239   1.367571675527904
       53    241   1.367595124169506
       54    251   1.367616745543125
       55    257   1.367637371380086
       56    263   1.367657068888028
       57    269   1.367675899358479
       58    271   1.367694453676628
       59    277   1.367712214558295
       60    281   1.367729474498914
       61    283    1.36774649198603
       62    293   1.367762369798154
       63    307   1.367776834902112
       64    311   1.367790931043951
       65    313    1.36780484804738
       66    317   1.367818416732261
       67    331   1.367830863665766
       68    337   1.367842872083792
       69    347   1.367854199412223
       70    349   1.367865397563672
       71    353   1.367876343814513
       72    359   1.367886927815935
       73    367    1.36789705611033
       74    373   1.367906861683412
       75    379   1.367916359717447
       76    383   1.367925660715323
       77    389    1.36793467742919
       78    397   1.367943334914839
       79    401   1.367951820810392
       80    409   1.367959978433511
       81    419   1.367967751816727
       82    421   1.367975451649847
       83    431    1.36798279877245
       84    433   1.367990078296927
       85    439     1.3679971604572
       86    443   1.368004115479064
       87    449   1.368010886100769
       88    457   1.368017422038038
       89    461   1.368023845197845
       90    463   1.368030213075281
       91    467    1.36803647247888
       92    479   1.368042422533127
       93    487   1.368048178930704
       94    491   1.368053842038096
       95    499   1.368059325219828
       96    503   1.368064721648104
       97    509   1.368069991746034
       98    521   1.368075022117894
       99    523   1.368080014145036
      100    541   1.368084679825012
      101    547   1.368089243819327
      102    557   1.368093645565895
      103    563   1.368097954088014
      104    569   1.368102172316221
      105    571   1.368106361085073
      106    577   1.368110463279219
      107    587   1.368114427024492
      108    593   1.368118311042942
      109    599   1.368122117716039
      110    601   1.368125899127191
      111    607    1.36812960622286
      112    613   1.368133241172475
      113    617   1.368136829191655
      114    619   1.368140394090371
      115    631   1.368143824802105
      116    641   1.368147149396719
      117    643   1.368150453365764
      118    647   1.368153716647514
      119    653   1.368156920289412
      120    659   1.368160065911614
      121    661     1.3681631925486
      122    673   1.368166208768089
      123    677   1.368169189483393
      124    683   1.368172118103236
      125    691   1.368174979358429
      126    701   1.368177759625487
      127    709   1.368180477553454
      128    719   1.368183120461192
      129    727   1.368185705567682
      130    733     1.3681882485598
      131    739   1.368190750458364
      132    743    1.36819322551363
      133    751   1.368195648157951
      134    757   1.368198032579947
      135    761   1.368200392022097
      136    769   1.368202702664003
      137    773   1.368204989473512
      138    787   1.368207195700638
      139    797   1.368209346949519
      140    809   1.368211434894317
      141    811   1.368213512563185
      142    821   1.368215539960657
      143    823   1.368217557525362
      144    827   1.368219555634991
      145    829   1.368221544123889
      146    839    1.36822348552457
      147    853   1.368225363760477
      148    857   1.368227224516902
      149    859   1.368229076626203
      150    863   1.368230911618634
      151    877    1.36823268852805
      152    881   1.368234449350101
      153    883     1.3682362022114
      154    887   1.368237939310085
      155    907    1.36823960068823
      156    911   1.368241247518877
      157    919   1.368242865820017
      158    929   1.368244449489433
      159    937   1.368246006248006
      160    941   1.368247549808529
      161    947   1.368249073883611
      162    953   1.368250578839905
      163    967   1.368252040558626
      164    971   1.368253490266873
      165    977   1.368254922234323
      166    983   1.368256336784649
      167    991   1.368257728601664
      168    997   1.368259103726767
      169   1009   1.368260446355096
      170   1013   1.368261778407689
      171   1019   1.368263094828804
      172   1021    1.36826440610137
      173   1031   1.368265692073827
      174   1033   1.368266973075147
      175   1039   1.368268239332433
      176   1049   1.368269481575149
      177   1051   1.368270719097879
      178   1061    1.36827193341504
      179   1063    1.36827314317032
      180   1069   1.368274339391037
      181   1087   1.368275496341443
      182   1091   1.368276644828646
      183   1093   1.368277789119509
      184   1097   1.368278925085448
    jagy@phobeusjunior
------------------

     95325   1233983   1.368327220256154
     95326   1234001   1.368327219555773
     95327   1234003   1.368327218853617
     95328   1234039   1.368327218117918
     95329   1234049   1.368327217372326
     95330   1234063   1.368327216612427
     95331   1234067   1.368327215848337
     95332   1234099   1.368327215048994
     95333   1234109   1.368327214237957
     95334   1234117   1.368327213417315
     95335   1234133   1.368327212576764
     95336   1234147   1.368327211717984
     95337   1234187   1.368327210802473
     95338   1234231    1.36832720981522
     95339   1234237   1.368327208817303
     95340   1234241   1.368327207812148
     95341   1234243   1.368327206803335
     95342   1234253   1.368327205775822
     95343   1234271    1.36832720471284
     95344   1234309   1.368327203566308
     95345   1234333   1.368327202359884
     95346   1234349    1.36832720110993
     95347   1234351   1.368327199854312
     95348   1234367   1.368327198551472
     95349   1234379   1.368327197210816
     95350   1234391   1.368327195830083
     95351   1234393   1.368327194442436
     95352   1234417   1.368327192966077
     95353   1234439   1.368327191397814
     95354   1234463   1.368327189715287
     95355   1234511   1.368327187745752
     95356   1234517   1.368327185733306
     95357   1234531    1.36832718361307
     95358   1234537   1.368327181443021
     95359   1234543   1.368327179220761
     95360   1234547   1.368327176962274
     95361   1234577   1.368327174389198
$\endgroup$
  • 1
    $\begingroup$ The exact value of the constant $\prod_{p} (p^2+p+1)/(p^2+p)$ is $\zeta(2)/\zeta(3) = 1.3684327776\ldots$ $\endgroup$ – Erick Wong Jan 10 '15 at 19:06
  • $\begingroup$ @ErickWong, I see. Good. en.wikipedia.org/wiki/… $\endgroup$ – Will Jagy Jan 10 '15 at 19:12
  • 1
    $\begingroup$ @Jose, given your interest, I suggest you learn Ramanujan's method in detail. It was introduced with the Superior Highly Composite Numbers. Later, Alaoglu and Erdos applied the same technique to make the Colossally Abundant Numbers. For real $\delta > 0,$ we would be finding the value of $n$ that maximises $$ \frac{\sigma(n^2)}{n^{1 + \delta} \sigma(n)};$$ my conjecture is that the best, call it $n_\delta,$ must be a primorial. $\endgroup$ – Will Jagy Feb 7 '15 at 23:26
  • 1
    $\begingroup$ Thank you for the references, @WillJagy! I'll go ahead and scrutinize Ramanujan's method. $\endgroup$ – Jose Arnaldo Bebita-Dris Feb 7 '15 at 23:29
  • 1
    $\begingroup$ @JoseArnaldoDris, note that those were not article titles, but asking wikipedia about them will lead you to the orrect references (other than my many answers using SHC or CA numbers). See also Hardy and Wright, Theorem 316, pages 260-261 in my edition, as a necessary step is to show that the function in question (including $\delta > 0$) goes to zero as $n$ goes to infinity. $\endgroup$ – Will Jagy Feb 7 '15 at 23:42

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