ASYMPTOTIC GCD AND DIVISIBLE SEQUENCES FOR ENTIRE FUNCTIONS

Let f and g be algebraically independent entire functions. We first give an estimate of the Nevanlinna counting function for the common zeros of f(n) - 1 and g(n) - 1 for sufficiently large n. We then apply this estimate to study divisible sequences in the sense that f(n) - 1 is divisible by g(n) - 1 for infinitely many n. For the first part of establishing our gcd estimate, we need to formulate a truncated second main theorem for effective divisors by modifying a theorem from a paper by Hussein and Ru and explicitly computing the constants involved for a blowup of P-1 x P-1 along a point with its canonical divisor and the pull-back of vertical and horizontal divisors of P-1 x P-1.