Why isn't a game purchased through Steam mobile showing in my PC library?

Solution 1:

When you purchase a game on Steam you will receive a purchase receipt in your Steam account's email Inbox. It looks similar to this:

If you do not have a receipt for your recent purchase in that Inbox, then you did not purchase the game with that account. Check your other Steam account's email addresses (if any) for the purchase receipt there.

If you did accidentally purchase the game with the wrong account and you haven't installed/played it yet, You should be able to request a refund from Steam then purchase it using the correct account. As per the image above, links to refund purchases are sent as part of the receipt.