Relative primes and transcendentals

 At the maximum expression for N relative primes the we get the  best estimate of Phi.

The estimate is N/(/N-1)  Exponent to an integer fraction will estimate Phi. Thus, any lesser irrational is estimated to best fractional approximation.

And a little manipulation sees how this might work with Pi when Phi -becomes log(phi,pi).

The best estimate of Pi as a N/(N-1) to some factional.  So, just estimate Phi itself and we get 109 = (01 + 17)/91, oddly.  There are an extra 17 combinations to sum up to maximum extent.

So finding the fraction estimate of Pi, for example.

Pi must be the best rational; option, and it a ratio of powers of Phi. Its exponential that makes Phi again must force the log Phi to an exponentiation which leaves Phi to the power of one.  But we now Phi to its maximum etant is we say in one dimension.  The other side of Markov has to make Phi again, but you have inserted Pi, the ongoing estimate.  You have translated the conditions into engineering units because Pi is the transcendental for Pi r^2, and you have put those conditions into the left side.   It is the same with finding the best fractional for e, euler's number.  If you insert the conditions o e in the left side of the equation,m then, keeping dimension, there is a search path.