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.
References
H. Cohen, $q$-identities for Maass waveforms, Invent. Math. 91 (1988), no. 3, 409--422.
Mathematical Reviews (MathSciNet):
MR928490
O. Espinosa and V. H. Moll, On some integrals involving the Hurwitz zeta function. I, Ramanujan J. 6 (2002), no. 2, 159--188.
O. Espinosa and V. H. Moll, On some integrals involving the Hurwitz zeta function. II, Ramanujan J. 6 (2002), no. 4, 449--468.
M. Hashimoto, S. Kanemitsu and M. Toda, On Gauss' formula for $\psi$ and finite expressions for the $L$-series at 1, J. Math. Soc. Japan 60 (2008), no. 1, 219--236.
S. Kanemitsu, J, Ma and W.-P. Zhang, On the discrete mean value of the product of two Dirichlet L-functions, Abh. Math. Sem. Univ. Hamburg. (to appear).
S. Kanemitsu, Y. Tanigawa and J. Zhang, Evaluation of Spannenintegrals of the product of zeta-functions, Integral Transforms Spec. Funct. 19 (2008), no. 1-2, 115--128.
S. Kanemitsu and H. Tsukada, Vistas of special functions, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007.
H. Liu and W. Zhang, On the mean value of $L(m,\chi)L(n,\overline\chi)$ at positive integers $m,n\geq 1$, Acta Arith. 122 (2006), no. 1, 51--56.
S. Louboutin, Quelques formules exactes pour des moyennes de fonctions $L$ de Dirichlet, Canad. Math. Bull. 36 (1993), no. 2, 190--196.
S. Louboutin, On the mean value of $\vert L(1,\chi)\vert \sp 2$ for odd primitive Dirichlet characters, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), no. 7, 143--145.
S. Louboutin, The mean value of $\vert L(k,\chi)\vert \sp 2$ at positive rational integers $k\geq1$, Colloq. Math. 90 (2001), no. 1, 69--76.
M. Mikolás, Mellinsche Transformation und Orthogonalität bei $\zeta(s,u)$. Verallgemeinrung der Riemannschen Funktionalgleichung von $\zeta(s)$, Acta Sci. Math. Szeged 17 (1956), 143--164.
Mathematical Reviews (MathSciNet):
MR89864
M. Mikolás, Integral formulae of arithmetical characteristics relating to the zeta-function of Hurwitz, Publ. Math. Debrecen 5 (1957), 44--53.
Mathematical Reviews (MathSciNet):
MR89865
L. J. Mordell, Integral formulae of arithmetical character, J. London Math. Soc. 33 (1958), 371--375.
Mathematical Reviews (MathSciNet):
MR100052
N. P. Romanoff, Hilbert spaces and number theory II, Izv. Akad. Nauk SSSR, Ser. Mat. 15 (1951), 131--152.
H. Walum, An exact formula for an average of $L$-series, Illinois J. Math. 26 (1982), no. 1, 1--3.
Mathematical Reviews (MathSciNet):
MR638548
A. Wintner, Diophantine approximations and Hilbert's space, Amer. J. Math. 66 (1944), 564--578.
Mathematical Reviews (MathSciNet):
MR11497
Y. Yamamoto, Dirichlet series with periodic coefficients, in Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976), 275--289, Japan Soc. Promotion Sci., Tokyo.
Mathematical Reviews (MathSciNet):
MR450213
