Improvement on $\phi(n)\sigma(n)/n^2$ bounds?

Question

We have:

$$\dfrac{6}{\pi^2}\lt\dfrac{\phi(n)\sigma(n)}{n^2}\le1$$

with equality iff $n=1$.

Wikipedia

Are there any known improvements on these bounds?

APPENDUM

For $n$ prime, $\dfrac{\phi\sigma}{n^2}\to1$.

Generally if $n=\prod p_i^{k_i}$, $\dfrac{\phi\sigma}{n^2}=\prod \big(1-\dfrac{1}{p_i^{k_i+1}}\big)$, which means the lower bound is sharp.

Answer 1

Call your function $f$.

Hint 1: If $n$ and $m$ are relatively prime, what is $f(nm)$?

Hint 2: If $n=p_1p_2\ldots p_k$ is a product of primes, what is $f(n)$? Does this expression remind you of anything?

Answer 2

I wrote a little program that displays your ratio for the primorial numbers, those are the products of consecutive primes beginning with $2;$ in the left of each line is that largest prime divisor, then the primorial itself, at the far right the ratio as a decimal. This is always larger than $6 / \pi^2 \approx 0.6079271.$ I had it keep printing until the ratio $\phi(P) \sigma(P) / P^2,$ for primorial $P,$ became smaller then $0.61.$ This happened when the largest prime factor of the primorial $P$ reached $59.$

1 1  phi  1  sigma  1 decimal 1
2 2  phi  1  sigma  3 decimal 0.75
3 6  phi  2  sigma  12 decimal 0.6666666666666666
5 30  phi  8  sigma  72 decimal 0.6399999999999999
7 210  phi  48  sigma  576 decimal 0.626938775510204
11 2310  phi  480  sigma  6912 decimal 0.6217574633159049
13 30030  phi  5760  sigma  96768 decimal 0.618078425071432
17 510510  phi  92160  sigma  1741824 decimal 0.6159397453999046
19 9699690  phi  1658880  sigma  34836480 decimal 0.6142335411190184
23 223092870  phi  36495360  sigma  836075520 decimal 0.6130724191131224
29 6469693230  phi  1021870080  sigma  25082265600 decimal 0.6123434388288024
31 200560490130  phi  30656102400  sigma  802632499200 decimal 0.6117062448237776
37 7420738134810  phi  1103619686400  sigma  30500034969600 decimal 0.6112594177640086
41 304250263527210  phi  44144787456000  sigma  1281001468723200 decimal 0.6108957893179859
43 13082761331670030  phi  1854081073152000  sigma  56364064623820800 decimal 0.6105653967872569
47 614889782588491410  phi  85287729364992000  sigma  2705475101943398400 decimal 0.6102889977846371
53 32589158477190044730  phi  4434961926979584000  sigma  146095655504943513600 decimal 0.6100717357704738
59 1922760350154212639070  phi  257227791764815872000  sigma  8765739330296610816000 decimal 0.6098964781618066

The other answer refers to Euler's identity for the Riemann Zeta function, which shows that the limit of the calculation below really is $1 / \zeta(2) = 6 / \pi^2.$ See https://en.wikipedia.org/wiki/Riemann_zeta_function#Euler_product_formula

I decided to compress the printout so as to allow larger numbers; here I give only the largest prime factor $p$ of $P$ and your ratio as a decimal. The ratio first becomes smaller than $0.61$ when $p = 59.$ The ratio first becomes smaller than $0.608$ when $p = 1049.$ The ratio first becomes smaller than $0.60793$ when $p = 19433.$ The limit is $6/\pi^2 \approx 0.607927101854.$

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     1   decimal   1
     2   decimal   0.75
     3   decimal   0.6666666666666666
     5   decimal   0.6399999999999999
     7   decimal   0.626938775510204
    11   decimal   0.6217574633159049
    13   decimal   0.618078425071432
    17   decimal   0.6159397453999046
    19   decimal   0.6142335411190184
    23   decimal   0.6130724191131224
    29   decimal   0.6123434388288024
    31   decimal   0.6117062448237776
    37   decimal   0.6112594177640086
    41   decimal   0.6108957893179859
    43   decimal   0.6105653967872569
    47   decimal   0.6102889977846371
    53   decimal   0.6100717357704738
    59   decimal   0.6098964781618066
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
   991   decimal   0.6080049189755558
   997   decimal   0.6080043073061252
  1009   decimal   0.6080037100999036
  1013   decimal   0.6080031176012898
  1019   decimal   0.6080025320601171
  1021   decimal   0.6080019488112501
  1031   decimal   0.6080013768222978
  1033   decimal   0.6080008070466026
  1039   decimal   0.6080002438330967
  1049   decimal   0.6079996913070232
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
 19387   decimal   0.6079300113127221
 19391   decimal   0.6079300096959337
 19403   decimal   0.6079300080811446
 19417   decimal   0.6079300064686832
 19421   decimal   0.607930004856886
 19423   decimal   0.6079300032454207
 19427   decimal   0.607930001634619
 19429   decimal   0.6079300000241488
 19433   decimal   0.6079299984143416
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=