Using ln as our utility, at time N we get:

  E[ln(X_1*...*X_N)] = sum E[ln(X)]

So kelly happens to maximize this expected utility by construction since it was derived by maximizing expected log of one round of betting (E[ln(X)]).

But if we use square root as our utility instead:

  E[sqrt(X_1*...*X_N)] = prod E[sqrt(X)]

We would maximize this expected utility by maximizing E[sqrt(X)]. Going through the same calculus we can see that we don't arrive at kelly.

Where did I go wrong?