"I am reading the newspaper."
you use ο/η/το when the noun is in τηε nominative case. Τhe noun is in its nominative case when it is the subject of the verb. In this case the noun (newspaper) is the object of the verb and in Greek it should be in its accusative case; when the noun is in the accusative case, its article must match its case; then you use τον/την/το.
In general Greek uses article, but there are many times that the article can be omitted. It is not mandatory, but Greek speakers do this mainly for brevity reasons, because the meaning doesn't change.
- Διαβάζω την εφημερίδα is correct too.
- Διαβάζω μία εφημερίδα is generally correct, but it is wrong in this sentence, because the English translation uses the article the and not a / one
don't be discouraged! you are correct! the proper translation of the sentence "I am reading the newspaper" is "διαβάζω την εφημερίδα". In English you are using the present continuous designating that you are reading the paper as you speak, therefore in Greek you need the article "την". On the other hand, if you say "Every morning I read the newspaper to kill my time", you can (but don't have to) omit the article in Greek: "κάθε πρωί διαβάζω (την) εφημερίδα για να σκοτώσω την ώρα μου"
Your are welcome, but not to confuse you I thought more about it and let me correct myself! The present continuous tense does not require the article! You may be asked: "Τί κάνεις;" = "What are you doing (at this moment)?" and you may answer "διαβάζω εφημερίδα", or "παίζω μπάλα" etc. I am not sure what the rule is, but that's the way it is!