I agree with you and your tutor – but mostly with you.

I agree with your tutor, because I'm not sure I would declare:

How is the number 5 written in the Roman numerals?

to be incorrect.

However, "not incorrect" is not the same as "usual", "idiomatic", or "common".

While both may be possible, the zero article is clearly better, in that it's less awkward and far more prevalent, as you discovered in your research:

How is the number 5 written in Roman numerals?

I'd also mention that we would need the definite article if we worded the sentence like this:

How is the number 5 written in the Roman numeral system?

In short, I wouldn't say that both options are equally valid, but I wouldn't go so far as to say the definite article is absolutely incorrect.

I agree with you and your tutor – but mostly with you.

I agree with your tutor, because I'm not sure I would declare:

How is the number 5 written in the Roman numerals?

to be incorrect.

However, "not incorrect" is not the same as "usual", "idiomatic", or "common".

While both may be possible, the zero article is clearly better, in that it's less awkward and far more prevalent, as you discovered in your research:

How is the number 5 written in Roman numerals?

I'd also mention that we would need the definite article if we worded the sentence like this:

How is the number 5 written in the Roman numeral system?

In short, I wouldn't say that both options are equally valid, but I wouldn't go so far as to say the definite article is absolutely incorrect. Just because you could include the article doesn't mean you should include the article.