MANIFESTATIONS OF THE GENERAL MODULAR RELATION

Author:

Fuhuo~LI,~Nianliang~WANG,~Shigeru~KANEMITSU

2010 Mathematics Subject Classification:

11F20, 11F32, 11J91, 14J15, 40C15.

Keywords:

Dedekind sum, eta-transformation, Hardy-Little-wood sum, manifestation of modular relation,  Meijer {tex inline}G{tex}-function, pseudo-
modular relation

Abstract:

In this paper, we shall exhibit a few instances of the manifestation of the general modular relation. The first is the Hardy-Littlewood sum studied by Segal, Kano and Berndt. We shall give a direct proof of Segal's main formula, locating it as a manifestation of the modular relation G_{0,1}^{1,0}\leftrightarrow G_{0,2}^{1,0}, where G stands for the Meijer G-function. We also state some close relationship to Koshlyakov's results. We can also give a partial answer as to whether there is a connection to the circle problem from the point of view of the gamma factor involved as a processing factor, the functional equation being not of Hecke's type. The second example is concerned with the reciprocity relation for the Dedekind sum. We shall show that it follows from the functional equation for the Riemann zeta-function in the long run via the well-known classical eta-transformation formula by way of the Hecke correspondence. The third example is a Lambert series considered by Wintner with respect to Riemann's posthumous fragment. It satisfies a pseudo-modular relation.

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Li-Wang-Kane-2012

Vol. 7 (15), 2011