Upper bound for the difference between number-of-divisors and sum-of-divisors functions

Question

The number-of-divisors function $d$, and the sum-of-divisors function $\sigma$, are defined by $$ d(n) = \sum_{d \mid n} 1, $$ $$ \sigma(n) = \sum_{d \mid n} d, $$ respectively. Now let $N$ be a square-free positive integer and consider the difference $$ d(N) - \dfrac{\sigma(N)}{N}. $$ Is there any kind of smooth function giving an upper bound for this?

Answer 1

for primorial $$N = \prod_{p \leq x} p,$$ we find $$ d(N) = 2^{\pi(x)} \approx 2^{x/\log x} $$ which is quite large, while $$ \sigma(N)/N = \prod_{p \leq x} 1 + \frac{1}{p} \approx \frac{6 e^\gamma \log x}{ \pi^2}, $$ much smaller. If the difference is not always an upper bound it will do until the real thing comes along.

Try it on computer. Note that i have written in terms of $x$ rather than writing some $p_n;$ that tends to create or erase logs.

Below is the result of the first 100 primes, the second column is the prime, the third column is the cumulative product of $1 + (1/p).$

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    1      2   1.5
    2      3   2
    3      5   2.4
    4      7   2.742857142857143
    5     11   2.992207792207792
    6     13   3.222377622377623
    7     17   3.411929247223365
    8     19   3.591504470761437
    9     23   3.747656839055412
   10     29   3.876886385229737
   11     31   4.00194723636618
   12     37   4.110107972484185
   13     41   4.210354508398433
   14     43   4.308269729523978
   15     47   4.399935042918106
   16     53   4.482952685237315
   17     59   4.558934934139642
   18     61   4.633671572404226
   19     67   4.702830849604289
   20     71   4.769067903824068
   21     73   4.834397601136726
   22     79   4.895592507480229
   23     83   4.954575549739027
   24     89   5.010244937938342
   25     97   5.06189694760781
   26    101   5.112014739168283
   27    103   5.16164595022817
   28    107   5.209885632006003
   29    109   5.257682747895966
   30    113   5.30421091380655
   31    127   5.345976354072743
   32    131   5.386785333874825
   33    137   5.426104934852014
   34    139   5.465141661002028
   35    149   5.501820464096002
   36    151   5.53825636120922
   37    157   5.573531879433482
   38    163   5.607725326546571
   39    167   5.641304520118706
   40    173   5.673913216766791
   41    179   5.705611055966606
   42    181   5.737133768982996
   43    191   5.767171118558823
   44    193   5.797052834199024
   45    197   5.826479498332014
   46    199   5.855758289780918
   47    211   5.883510698737226
   48    223   5.909894154785374
   49    227   5.935928930797645
   50    229   5.961850017831695
   51    233   5.98743735696402
   52    239   6.012489396114497
   53    241   6.037437484895055
   54    251   6.06149102069145
   55    257   6.085076588865347
   56    263   6.10821376220704
   57    269   6.130920876564687
   58    271   6.153544200832454
   59    277   6.175759161846289
   60    281   6.197736952457842
   61    283   6.219637083031898
   62    293   6.240864513349413
   63    307   6.261193062252831
   64    311   6.281325515829207
   65    313   6.301393648467639
   66    317   6.321271861869745
   67    331   6.340369359941858
   68    337   6.359183512345247
   69    347   6.377509689614254
   70    349   6.395783356346673
   71    353   6.413901722795247
   72    359   6.431767744307211
   73    367   6.449292997016495
   74    373   6.46658332676721
   75    379   6.483645551903798
   76    383   6.500574130368299
   77    389   6.517285117849966
   78    397   6.533701453159411
   79    401   6.549994972992726
   80    409   6.566009630628406
   81    419   6.581680298004607
   82    421   6.597313742892979
   83    431   6.612620735335886
   84    433   6.627892376756985
   85    439   6.642990081487638
   86    443   6.657985544425533
   87    449   6.67281402002559
   88    457   6.687415363614267
   89    461   6.701921687613431
   90    463   6.716396680459248
   91    467   6.730778686198989
   92    479   6.744830416232808
   93    487   6.758680170680925
   94    491   6.772445303411436
   95    499   6.786017338087611
   96    503   6.799508426234903
   97    509   6.81286698895835
   98    521   6.825943509090708
   99    523   6.838995026316502
  100    541   6.851636421928918

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Answer 2

The number of divisors is usually so small that it doesn't make a difference, so the records are usually the highly abundant numbers, A002093 in the OEIS.

As an example, the $10^4$-th highly abundant number is $N=7442466942548913946301030400=2^{13}\cdot3^4\cdot5^2\cdot7^2\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31\cdot37\cdot41\cdot43\cdot47\cdot53\cdot59$ which has 5160960 divisors summing to 53306653535048357847760896000.

If you restrict to squarefree numbers only, then Will's answer is spot on: you can't do better than primorials.