Explicit Formulas and Asymptotic Expansions for Certain Mean Square of Hurwitz Zeta-Functions: III

• Published in: Compositio mathematica Vol. 131; no. 3; pp. 239 - 266
• Main Authors: Katsurada, Masanori, Matsumoto, Kohji
• Format: Journal Article
• Language: English
• Published: London, UK London Mathematical Society 01.05.2002; Cambridge University Press
• Subjects: Riemann zeta function; mean square; Hurwitz zeta function; asymptotic expansion
• ISSN: 0010-437X; 1570-5846
• DOI: 10.1023/A:1015585314625

The main object of this paper is the mean square Ih(s) of higher derivatives of Hurwitz zeta functions ζ(s, α). We shall prove asymptotic formulas for Ih(1/2 + it) as t → +∞ with the coefficients in closed expressions (Theorem 1). We also prove a certain explicit formula for Ih(1/2 + it) (Theorem 2), in which the coefficients are, in a sense, not explicit. However, one merit of this formula is that it contains sufficient information for obtaining the complete asymptotic expansion for Ih(1/2 + it) when h is small. Another merit is that Theorem 1 can be strengthened with the aid of Theorem 2 (see Theorem 3). The fundamental method for the proofs is Atkinson's dissection argument applied to the product ζ(u, α)ζ(v, α) with the independent complex variables u and v.