  One familiar with the Euler zeta function, which established the remarkable
relationship between the prime and composite numbers, might naturally ponder
the results of the application of this special function in cases where there is
no known way to sieve composite numbers out of the product term in this famous
equation. Such would be case when an infinite series of numbers to be analyzed
are calculated by a polynomial expression that yields successively increasing
positive integer values and which has as its input domain the positive integers
themselves. In such cases there may not be an intuitive way to eliminate the
composite terms from the product term on the right-hand side of the Euler zeta
function equation by either scaling a previous prime number calculation or by
employing predictable values of the domain of the function which would make
future outputs of the polynomial prime. So the best one may be able to hope for
in this case is to determine some value to be added or subtracted from unity in
the numerator above the product term to make both sides of the equation equal
in the hope that that value can be used to predict the number of prime numbers
that exist as outputs of the polynomial function for some limit less than or
equal to x of the input domain.
