Structural elucidation of the mean square of the Hurwitz zeta-function

In this note we shall give a complete structural description of the mean square of the Hurwitz zeta-function whose study was started 50 years ago. Instead of appealing to Atkinson's dissection, we incorporate the built-in structure of the Hurwitz zeta-function as the solution of the difference equation. First we shall prove a Katsurada–Matsumoto formula from which the best asymptotic expansion for the mean square at follows by K-times integration by parts, and then we shall show that their fundamental formula is essentially the N-times integration by parts of the same formula. The key is to introduce a suitable function fκ the integration of which gives , and then to view as .