Source: J. Math. Soc. Japan Volume 61, Number 3
(2009), 651-660.
Espinosa and Moll [2], [3] studied “the Hurwitz transform” meaning an integral over [0, 1] of a Fourier series multiplied by the Hurwitz zeta function $\zeta (z,u)$, and obtained numerous results for those which arise from the Hurwitz formula. Ito's recent result [4] turns out to be one of the special cases of Espinosa and Moll's theorem. However, they did not give rigorous treatment of the relevant improper integrals.
In this note we shall appeal to a deeper result of Mikolás [9] concerning the integral of the product of two Hurwitz zeta functions and derive all important results of Espinosa and Moll. More importantly, we shall record the hidden and often overlooked fact that some novel-looking results are often the result of “duplicate use of the functional equation”, which ends up with a disguised form of the original, as in the case of Johnson's formula [5]. Typically, Example 9.1 ((1.12) below) is the result of a triplicate use because it depends not only on our Theorem 1, which is the result of a duplicate use, but also on (1.3), the functional equation itself.
References
R. Balasubramanian, L.-P. Ding, S. Kanemitsu and Y. Tanigawa, On the partial fraction expansion for the cotangent-like function, Proc. Chandigarh Conf., Ramanujan Math. Soc. Lect. Notes Ser., 6, Ramanujan Math. Soc. Mysore, 2008, pp. 19–34.
O. Espinosa and C. Moll, On some integrals involving the Hurwitz zeta-function: Part 1, Ramanujan J., 6 (2002), 159–188.
O. Espinosa and C. Moll, On some integrals involving the Hurwitz zeta-function: Part 2, Ramanujan J., 6 (2002), 449–468.
T. Ito, On an integral representation of special values of the zeta function at odd integers, J. Math. Soc. Japan, 58 (2006), 681–691.
S. Kanemitsu, Y. Tanigawa and H. Tsukada, A generalization of Bochner's formula, Hardy-Ramanujan J., 27 (2004), 28–46.
S. Kanemitsu, Y. Tanigawa, H. Tsukada and M. Yoshimoto, Contributions to the theory of the Hurwitz zeta-function, Hardy-Ramanujan J., 30 (2007), 31–55.
S. Kanemitsu, Y. Tanigawa and J.-H. Zhang, Evaluation of Spannenintegrals of the product of zeta-functions, Integral Transforms Spec. Funct., 19 (2008), 115–128.
S. Kanemitsu and H. Tsukada, Vistas of special functions, World Scientific, Singapore etc. 2006.
M. Mikolás, Mellinsche Transformation und Orthogonalität bei $\zeta(s,u)$. Verallgemeinerung der Riemannschen Funktionalgleichung von $\zeta(s)$, Acta Sci. Math. Szeged, 17 (1956), 143–164.
Mathematical Reviews (MathSciNet):
MR89864
M. Mikolás, A simple proof of the functional equation for the Riemann zeta-function and a formula of Hurwitz, Acta Sci. Math. Szeged, 18 (1957), 261–263.
Mathematical Reviews (MathSciNet):
MR93500
Y. Yamamoto, Dirichlet series with periodic coefficients, Algebraic Number Theory, Proc. Internat. Sympos, Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976, Japan. Soc. Promotion Sci. Tokyo, 1977, pp. 275–289.
Mathematical Reviews (MathSciNet):
MR450213
