Proc. Japan Acad. Ser. A Math. Sci. 85 (9), 149-154, (November 2009) DOI: 10.3792/pjaa.85.149
Kalyan Chakraborty, Shigeru Kanemitsu, Jinhon Li, Xiaohan Wang
KEYWORDS: Parseval identity, orthnormal basis, Dirichlet L-function, 11R58, 11R29
In this paper, we make structural elucidation of some interesting arithmetical identities in the context of the Parseval identity.
In the continuous case, following Romanoff [R] and Wintner [Wi], we study the Hilbert space of square-integrable functions L2(0,1) and provide a new complete orthonormal basis-the Clausen system-, which gives rise to a large number of intriguing arithmetical identities as manifestations of the Parseval identity. Especially, we shall refer to the identity of Mikolás-Mordell.
Secondly, we give a new look at enormous number of elementary mean square identities in number theory, including H. Walum's identity [Wa] and Mikolás' identity (1.16). We show that some of them may be viewed as the Parseval identity. Especially, the mean square formula for the Dirichlet L-function at 1 is nothing but the Parseval identity with respect to an orthonormal basis constructed by Y. Yamamoto [Y] for the linear space of all complex-valued periodic functions.