MANIFESTATIONS OF THE GENERAL MODULAR RELATION
Author:


2010 Mathematics Subject Classification:

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

Keywords:

Dedekind sum, etatransformation, HardyLittlewood 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 HardyLittlewood 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 , where stands for the Meijer 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 zetafunction in the long run via the wellknown classical etatransformation 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 pseudomodular relation.

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Vol. 7 (15), 2011
