Question

euclid :: Int -> Int
euclid n = length (filter (gcd n == 1) [1 .. n-1])

gcd :: Int -> Int -> Int

..

Answer 1

Assuming you’re looking for Euler’s totient function, simply call

euler_fi1 n = length $ filter ((==1).(gcd n)) [1..n-1]

The linked WP article gives a formula for calculating this directly:

euler_fi n = let 
   fs = Data.List.nub $ factorize n 
   pr = n * product [p-1 | p <- fs] 
  in Data.List.foldl' div pr fs

You’ll need a factorize function for that:

factorize n | n > 1 = go n (2:[3,5..])  where
  go n ds@(d:t)
    | d*d > n    = [n]
    | r == 0     =  d : go q ds
    | otherwise  =      go n t
        where 
          (q,r)  = quotRem n d

Next optimization is to use primes list instead of (2:[3,5..]).