# Improving the inequality $x\sigma_1(x) \leq \sigma_1(x^2)$ for $x \in \mathbb{N}$

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 $x$.

• Related MSE question – Jose Arnaldo Bebita-Dris Jan 9 '15 at 16:27
• 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. – Erick Wong Jan 11 '15 at 10:39
• Okay, thanks @ErickWong! – Jose Arnaldo Bebita-Dris Jan 12 '15 at 7:50

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

• The exact value of the constant $\prod_{p} (p^2+p+1)/(p^2+p)$ is $\zeta(2)/\zeta(3) = 1.3684327776\ldots$ – Erick Wong Jan 10 '15 at 19:06
• @ErickWong, I see. Good. en.wikipedia.org/wiki/… – Will Jagy Jan 10 '15 at 19:12
• @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. – Will Jagy Feb 7 '15 at 23:26
• Thank you for the references, @WillJagy! I'll go ahead and scrutinize Ramanujan's method. – Jose Arnaldo Bebita-Dris Feb 7 '15 at 23:29
• @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. – Will Jagy Feb 7 '15 at 23:42