By: Tao Steven Zheng (鄭濤)

【Problem】

Here are three infinite series problems I created that involves the Zeta function: .

Problem 1: Show that .

Problem 2: GIven that , calculate the series and .

Problem 3: Compute the infinite series .

【Solution】

Problem 1

Problem 2

（1）Calculate .
From problem 1:

Since , then

（2）Calculate .

Since , we have

Thus,

Problem 3

Therefore,