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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A zeta-function associated with zero ternary forms


Author: Min King Eie
Journal: Proc. Amer. Math. Soc. 94 (1985), 387-392
MSC: Primary 11E45; Secondary 11F99, 11M41
DOI: https://doi.org/10.1090/S0002-9939-1985-0787878-9
MathSciNet review: 787878
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Abstract: Consider the zeta-function associated with zero ternary forms defined as

$\displaystyle \tilde \xi (t) = \sum\limits_x {\frac{1} {{{{\left\vert {\det x} \right\vert}^t}}}} \quad (\operatorname{Re} t \geqslant 2),$

where $ x$ runs over all $ \operatorname{SL}_{3}({\mathbf{Z}})$-inequivalent zero ternary forms. We shall approximate $ \tilde \xi (t)$ by another zeta-function which we can compute explicitly. By the approximation, we see that $ \tilde \xi (2)$ is very close to $ 2\zeta (2)\zeta (2)$ which gives the contribution of zero ternary forms to the dimension formula of Siegel's cusp forms of degree three (computing via Selberg Trace Formula) up to a constant multiple.

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DOI: https://doi.org/10.1090/S0002-9939-1985-0787878-9
Article copyright: © Copyright 1985 American Mathematical Society