Primes less a given number n (n >= 2) determines new primes within a limited area increased with a square (n2) or decreased with a square root (sqrt(��)). As the area is extended, the number of primes is also changed and controlled within an extended area boundary or number boundary, n to n2 or n to sqrt(��). The structure of a number boundary is applied to the Euler product and helps to characterize the Euler’s prime boundary between n and (n2 - 1). The characterized Euler product is used to characterize the non-trivial zeroes derived in an elementary way of Riemann zeta function. Then, the characterized Euler product and non-trivial zeroes are discussed regarding their potential number boundaries. Overall, it is concluded that the characteristic of a number boundary can represent the characteristic of primes, especially the number of primes. As the number boundary is characterized by the increased or decreased exponent while the base or given number n is fixed, it is concluded that the pattern of exponent in the number boundary would be a key to understanding the pattern of primes.