Manifestations of the Parseval identity

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 L-2(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 Mikolas-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 Mikolas' 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.