Please read the comments especially if you run into problems. They contain some valuable information which I can't integrate into the guide because of lack of time to test it. Also GMail has limit of 100 messages/day/account when you use SMTP. Keep this in mind :)
This blog runs on wordpress and the "comment notifications by email" never worked. Initially I thought it was a wordpress problem and trawled through forum posts looking for a solution. Unable to find one I did something which I should have tried right at the beginning. I tried sending a mail from the server using the 'mail' command. Oops !. Exim log(/var/log/exim4/mainlog) give the wonderfully worded message R=nonlocal: Mailing to remote domains not supported
Also I wanted to route the mail through GMail SMTP servers since that would improve reliability. Turns out there is an indeed a way to make exim, the default MTA in debain/ubuntu to route all mail through GMail SMTP servers. Second round of trawling the interwebs started I managed to get a working setup without much trouble. But I had to piece together the content from separate sources. Here is how I did it.
Run dpkg-reconfigure exim4-config
send_via_gmail: driver = manualroute domains = ! +local_domains transport = gmail_smtp route_list = * smtp.gmail.comIf you have any other smarthost defined with "domains = ! +local_domains" remove that smarthost.
gmail_login: driver = plaintext public_name = LOGIN client_send = : email@example.com : YourGmailPasswordMake sure you have no other authenticators with the same public_name (LOGIN). Comment them out if needed (Thanks Jakub for reminding me)
gmail_smtp: driver = smtp port = 587 hosts_require_auth = $host_address hosts_require_tls = $host_address
That should be it. You can test by using the command line mail client.
This was on an Ubuntu server. I believe that this instructions will also work on Debain without any need for modifications.