Anyway, my idea was to construct a number which is a base 10 concatonation of the digits of consecutive prime numbers:
mp = 0.2357911131719232931...
or the natural numbers:
mN = 0.12345678910111213...
The construction proceeded as follows. First we define the function lambda:
Now we can express the numbers as infinite sums:
Initially I thought this to be an exercise in creative procrastination, since firstly the idea is dead simple, and secondly who would be silly enough in going to the trouble of formulating such numbers. Surely only a fresh graduate with too much time on his hands.
To my surprise the idea behind those numbers has been already formulated by not one, but a few mathematicians. My mN is in fact a constant previously formulated by D. G. Champernowne. And it turns out that mp is already known as the
Notice the striking similarity in the formulation - I have formulated mp completly oblivious to the Copeland-Edros formulation. The similarity lies in the fact that my lambda function acts a lot like the floor of base 10 logarithm of y, plus 1. In fact the relation is the following:
Which makes the sums in the exponents of the definitions of mp and the Copeland-Edros constant equal, since for each i we subtract 1 i.e. we subtract 1, k times, we need to add k back to the sum.
Well, I guess I feel better knowing that I'm in good company deeming quite obscure and prima facie pointless ideas valuable. I knew that both mN and mp are irrational, but now I also know that mN is in fact transcendental (mission accomplished!), and hence don't have to provide a proof, which i feel would be a herculean task.