@Op,
The thing you are missing is software. "Downloads" don't happen in a vaccuum; a piece of software requests a resource, by opening an IO stream to it. In general The Server responds by sending the requested resource as part of that stream. If the stream ends (on either side), the download is broken.
So, in the normal case, the PC will walk up to the counter and say "hey, I'm here to pickup that order..." and the server at the counter will say "ok, heres the first box; take it out to the car, and come back to get the next 6billion of them".
In your case, the PC walks up to the counter to pick up the order. they take the first box out to the car, but then drive away, and the server is just waiting for the PC to come back to get more.
So ultimately (dumb yet apt analogy aside) without software running locally to operate one end of the IO stream, no downloading will happen. TCP won;t just keep sending or recieving data when one of the parties to the stream disappears. when that happens, the whole stream breaks. Additionally in your example, the PC is sending the downloaded data to the remote hard disk, not the remote server, which it can't do when off.
As @Tyson suggested, one solution to this is to deploy software to the router itself. That is certainly possible. OpenWRT is one common firmware replacement. Feel free to investigate whether it will run the software necessary to perform whatever particular download operations you are looking for.
1What you describe is not plausible. Most firmware found on routers do not have the capability to mount external HDD's volume.. Some firmware found on routers allow you to attach the external HDD in such a way where it becomes a network drive. Most modem firmware is extremely locked down by your internet provide, so if the device you have is acting like a modem/gateway, then what you want is not plausible in the slightest. – Ramhound – 2015-12-29T19:11:28.223
@Ramhound but if I plug the hard drive to the router and I connect to it using the samba protocol I can actually download files on it; so are you saying there is not a workaround where I initiate the download through the PC and then I turn it off leaving the the router on and downloading the data on the external hard drive attached to it? – Federico Gentile – 2015-12-29T19:23:57.667
1The only "work around" I see is if you can load alternative firmware onto your router, and then script directly on the router as necessary to facilitate your download needs. google dd-wrt or tomato for more info. beware tho, it's not as straightforward as would seem. – Tyson – 2015-12-29T19:29:35.743
1
Ramhound! All of the routers that I have had that have had USB ports, DLink, Linksys, TPlink, Netgear and a couple others have had USB ports, and you could run a bittorrent client right on the device. It looks like the OP's router is this one: http://www.dlink.com/uk/en/support/product/dsl-2750b-wireless-n-adsl2-plus-modem-router#
– David – 2015-12-29T19:32:56.930@FedericoGentile, please could you link to webpage of your router. It looks like that it is a Dlink DSL-2750B router. – David – 2015-12-29T19:34:00.847
@Ramhound - I used to live in BC where Telus would provide you with locked down DSL modems, and you would connect your own router through the modem. I now live in Europe, like most of the world, the phone company does not provide you with a modem or router, you buy your own. – David – 2015-12-29T19:36:07.883
@FedericoGentile - Yes; What you describe in your comment is not plausible. – Ramhound – 2015-12-29T21:34:11.400