Discrete fundamental region and its asymptotics
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The calculus presented here may be called multiplicative in view of an essential dependence of it on the multiplicative structure of the function ζ D (s) from the equality (3.1).
But at the same time, the function ζ D (s) possesses also an additive structure of the form of the sum over the roots ω j from Lemma 1.1. Therefore, it is possible to calculate the square mean (3.2) using the sums of Kloosterman-Salié sums in the average with respect toD, and to obtain its asymptotics by a spectral approach using Kuznetsov's formulas (see ) for summing on Weil's metaplectic group . The possibility of connecting the Riemann hypothesis with the square mean estimates of the discrete spectrum of the metaplectic group appears in this way.
KeywordsDiscrete Spectrum Additive Structure Riemann Hypothesis Fundamental Region Spectral Approach
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