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Journal für die reine und angewandte Mathematik

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Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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A new approach to the spectral theory of the fourth moment of the Riemann zeta-function

Roelof W. Bruggeman1 / Yoichi Motohashi2



Citation Information: Journal für die reine und angewandte Mathematik. Volume 2005, Issue 579, Pages 75–114, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crll.2005.2005.579.75, July 2005

Publication History

5. Dezember 2003
Published Online:


The aim of the present article is to exhibit a new proof of the explicit formula for the fourth moment of the Riemann zeta-function that was established by the second named author a decade ago. Our proof is new, particularly in that it dispenses altogether with the spectral theory of sums of Kloosterman sums that played a predominant rôle in the former proof. Our argument is, instead, built directly upon the spectral structure of the space L 2(Γ\G), with Γ = PSL2(ℤ) and G = PSL2(ℝ). The discussion below thus seems to provide a new insight into the nature of the Riemann zeta-function, especially in its relation with automorphic forms over linear Lie groups that has been perceived by many.

The plan of the paper is as follows: In Section 1 we discuss salient points of the former proof and describe the explicit formula in a conventional fashion. In Section 2 its reformulation is presented in terms of automorphic representations occurring in L 2(Γ\G). With this, our motivation is precisely related. We then proceed to our new proof. In Section 3 we construct a Poincaré series over G whose value at the unit element is close to the nondiagonal part of the fourth moment in question. In Section 4 we develop an account of the Kirillov scheme, with which, in Section 5, projections of the Poincaré series into irreducible subspaces of L 2(Γ\G) are explicitly calculated in terms of the seed function. Then, in Section 6 a limiting procedure with respect to the seed is performed, and we reach a basic spectral expression. This ends in effect our proof of the explicit formula, since it remains to appeal to a process of analytic continuation, which is, however, the same as the corresponding part of the former proof, and can largely be omitted.

Notations are introduced where they are needed for the first time, and will continue to be effective thereafter. The parameters C > 0 and ε > 0 are assumed locally to be constants arbitrarily large and small, respectively. The dependency of implicit constants on them can be inferred from the context.

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