In the key of F major, what roman numeral is the F# A C chord? I know it is a diminished chord. It is going from a F to F#dim to Gm7
Using Roman numeral analysis you can look at it as a I - viio/ii - ii7 in the key of F, but I would look at it slightly differently see below.
Roman numeral analysis doesn't work in all cases and in fact I wouldn't give this chord a roman numeral since it is a chromatic passing chord between two chords in the key.
Think of it this way you are going from I to ii7. The F#dim is sandwiched in between the two chords and the F# can be perceived as a chromatic passing tone between the F and G. Because in the F# diminished the A and the C are common, the F#dim can be seen as an altered version of the I chord and thus the harmony from the analysis standpoint doesn't change.
If you really wanted to use a Roman numeral you could use a secondary dominant to describe the progression as a viio/ii making the whole progression I - viio/ii - ii7. I personally think just saying it is a passing chord is what makes most sense, but this is also a valid way to look at the progression.
It could be written as vo or iiio as both of those notes feature in the diminished chord. Could even be bviio as an Eb may also be played. However, as the lowest note is apparently an F#, it could also be written #Io,(#io) although I've never seen that written. The lower case seems to be used to indicate a 'minor' chord, as a diminished has a minor third.Couldn't find much about this while searching.
It MAY not be a diminished chord, though. It could be the dominant of that Gm, without the root. (D) F# A C. This would put the Roman numeral as VI7.Or, as Basstickler points out, V7/ii7.