If I remember right, there is an option to claim warranty online.
Contact Logitech. They may suggest you some troubleshooting options (depends on the issue). Even after troubleshooting still the issue not fixed, they will ask you to take it to the nearest service center. Yes, you will need to take bill with you.
In my case, it was a mouse. And they replaced it right away with a refurbished one.
Get the invoice from Amazon. Go to the service center with your keyboard and explain them the problem you are facing and show them the invoice copy in your smartphone. They should ask you to send the invoice copy to them over email and then process the warranty claim after checking the issue. Take printout only if they insist, save paper! I did same when I went to claim warranty for MK350 combo