TY - JOUR

T1 - Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions

T2 - III

AU - Katsurada, Masanori

AU - Matsumoto, Kohji

N1 - Funding Information:
M.K. was supported in part by Grant-in-Aid for Scienti¢c Research (N o. 11640038), Ministry of Education, Science, Sports and Culture of Japan.

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - Asymptotic expansion

KW - Hurwitz zeta function

KW - Mean square

KW - Riemann zeta function

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U2 - 10.1023/A:1015585314625

DO - 10.1023/A:1015585314625

M3 - Article

AN - SCOPUS:0036277265

SN - 0010-437X

VL - 131

SP - 239

EP - 266

JO - Compositio Mathematica

JF - Compositio Mathematica

IS - 3

ER -