Research On ζ Function And Complete Square Number

In this thesis, we study two sections as follows:In the first part we know that the Riemann ζ-function is a special form of the Hurwitz ζ-function, and there do exist many series that can represent the Riemann ζ-function. In this article, we prove that a convergent series can represent the Hurwitz ε-function. That is the theorem2.3.1.Theorem2.3.1If x∈R, α∈C C,0<|x|<2π,(?)n∈N+, Theorem2.3.1is from the theorem2.3.2.Theorem2.3.2If x∈R R,α∈C,0<|x|≤2π,(?)n∈N+In the second part, W. D.Banks and F. Luca have proved that there are infinitely many pairs (p,q) of distinct primes such that (p-1)(q-1)is a perfect square. In this article we get a new result.Theorem3.2. Let c≥1be a positive integer, there are infinitely many pairs (p, q) of distinct primes such that (p—c)(q—c) is a perfect square.