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Manifestations of the Parseval identity. (English) Zbl 1251.11060

Summary: We make structural elucidation of some interesting arithmetical identities in the context of the Parseval identity.
In the continuous case, following N. P. Romanov [Izv. Akad. Nauk SSSR, Ser. Mat. 10, 3–34 (1946); 15, 131–152 (1951; Zbl 0044.04002)] and A. Wintner [Am. J. Math. 66, 564-578 (1944; Zbl 0061.24902)], 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 Mikolás-Mordell.
Secondly, we give a new look at an enormous number of elementary mean square identities in number theory, including H. Walum’s identity [Ill. J. Math. 26, 1–3 (1982; Zbl 0464.10030)] and Mikolás’ identity. 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 [Algebr. Number Theory, Pap. Kyoto Int. Symp. 1976, 275–289 (1977; Zbl 0371.10028)] for the linear space of all complex-valued periodic functions.

MSC:

11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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