I will assume positive integers because or else we can maximize the product without bounds.
Lets say that there are y integers whose sum is 2008. Denote them by .
We want to maximize the product
Since by assumption the numbers are positive, we can apply AM-GM inequality.
Thus the maximum value of the product is and is purely determined by y.
So we want to maximize
A little bit of calculus establishes that f(y) approaches maxima at
Thus the discrete maximum is attained at (Actually I have done trial error to choose between 738 and 739)
In AM-GM inequality, equality holds when all the numbers are equal.
Thus I believe that is the largest product.