Large gap between two consecutive square-free numbers

Question

Let $q_n$ denote the $n$-th square-free number. By Chinese remainder theorem (see this post), it is not difficult to show that there is arbitrarily large gap between two consecutive square-free numbers, i.e., $\limsup_{n\to\infty}(q_{n+1}-q_n)=\infty$. How to prove the stronger bound? $$\limsup_{n\to\infty}\frac{q_{n+1}-q_n}{\log n/\log\log n}\geq \frac{1}{2}.$$ I don't how to get started, thanks for any help.

Edit: This is an exercise (Exercise 2.20) from A.J. Hildebrand's An introduction to analytic number theory.

Answer 1

I was curious about how large that ratio could stay. I had it print out only when the ratio exceeded 1.5. Does not seem to be dying out. The later part of that exercise says prove the limsup is at least $\pi^2/12 \approx 0.822467$ From what I am seeing, a limsup between 1 and 2 seems a good bet. Probably hard to prove, though.

First, just when the ratio increases:

 n = 4   old q  3  q 5  ratio  0.1712120437515166    gap  2
 n = 5   old q  5  q 6  ratio  0.2356168135275511    gap  1
 n = 6   old q  6  q 7  ratio  0.2956839724294714    gap  1
 n = 7   old q  7  q 10  ratio  0.9764671388072846    gap  3
 n = 17   old q  23  q 26  ratio  1.103425220291327    gap  3
 n = 32   old q  47  q 51  ratio  1.437072374330929    gap  4
 n = 151   old q  241  q 246  ratio  1.608142099431302    gap  5
 n = 516   old q  843  q 849  ratio  1.760024446369004    gap  6

Now, ratio larger than 1.5

 n = 151   old q  241  q 246  ratio  1.608142099431302    gap  5
 n = 516   old q  843  q 849  ratio  1.760024446369004    gap  6
 n = 1026   old q  1679  q 1685  ratio  1.675782937474057    gap  6
 n = 1756   old q  2887  q 2893  ratio  1.615152343691931    gap  6
 n = 2208   old q  3623  q 3629  ratio  1.590625635457252    gap  6
 n = 3071   old q  5045  q 5051  ratio  1.556608495338398    gap  6
 n = 13392   old q  22019  q 22026  ratio  1.658619700318701    gap  7
 n = 14985   old q  24646  q 24653  ratio  1.647791097638391    gap  7
 n = 18804   old q  30922  q 30929  ratio  1.626378938461883    gap  7
 n = 28995   old q  47671  q 47678  ratio  1.587166663694555    gap  7
 n = 33718   old q  55446  q 55453  ratio  1.573982134381624    gap  7
 n = 34728   old q  57119  q 57126  ratio  1.571431499337303    gap  7
 n = 44650   old q  73446  q 73453  ratio  1.550075102998279    gap  7
 n = 45501   old q  74847  q 74854  ratio  1.54849635050823    gap  7
 n = 58776   old q  96674  q 96681  ratio  1.527433347355694    gap  7
 n = 64310   old q  105771  q 105778  ratio  1.520178814511105    gap  7
 n = 73964   old q  121666  q 121673  ratio  1.509052248661901    gap  7
 n = 74071   old q  121846  q 121853  ratio  1.50893818425337    gap  7
 n = 131965   old q  217069  q 217077  ratio  1.674107764035864    gap  8
 n = 408142   old q  671345  q 671353  ratio  1.584428949113135    gap  8
 n = 502653   old q  826823  q 826831  ratio  1.569036523505159    gap  8
 n = 664314   old q  1092746  q 1092755  ratio  1.742561268994944    gap  9
 n = 867735   old q  1427369  q 1427377  ratio  1.530225943242189    gap  8
 n = 1274859   old q  2097047  q 2097055  ratio  1.50414070948736    gap  8
 n = 4387186   old q  7216617  q 7216626  ratio  1.605004820261783    gap  9
 n = 5392319   old q  8870023  q 8870033  ratio  1.768248678666427    gap  10
 n = 8741561   old q  14379270  q 14379279  ratio  1.560603965676062    gap  9
 n = 13760650   old q  22635346  q 22635355  ratio  1.532852419427845    gap  9
 n = 15086919   old q  24816973  q 24816982  ratio  1.527358881812587    gap  9
 n = 15227282   old q  25047845  q 25047854  ratio  1.526808447544024    gap  9
^C

Worth noting that the actual $q_{n+1} - q_n$ are quite small, near the end of this output just 9 or 10. Eh, shorter output, I told it to keep going forever and just print when the gap size increases. Here it is so far:

 n = 4   old q  3  q 5  ratio  0.1712120437515166    gap  2
 n = 7   old q  7  q 10  ratio  0.9764671388072846    gap  3
 n = 32   old q  47  q 51  ratio  1.437072374330929    gap  4
 n = 151   old q  241  q 246  ratio  1.608142099431302    gap  5
 n = 516   old q  843  q 849  ratio  1.760024446369004    gap  6
 n = 13392   old q  22019  q 22026  ratio  1.658619700318701    gap  7
 n = 131965   old q  217069  q 217077  ratio  1.674107764035864    gap  8
 n = 664314   old q  1092746  q 1092755  ratio  1.742561268994944    gap  9
 n = 5392319   old q  8870023  q 8870033  ratio  1.768248678666427    gap  10

This last version is extended at https://oeis.org/A020754 and https://oeis.org/A020754/b020754.txt