A zeta-function associated with zero ternary forms
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- by Min King Eie PDF
- Proc. Amer. Math. Soc. 94 (1985), 387-392 Request permission
Abstract:
Consider the zeta-function associated with zero ternary forms defined as \[ \tilde \xi (t) = \sum \limits _x {\frac {1} {{{{\left | {\det x} \right |}^t}}}} \quad (\operatorname {Re} t \geqslant 2),\] 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.References
- Takuro Shintani, On zeta-functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo Sect. I A Math. 22 (1975), 25–65. MR 0384717
- C. L. Siegel, Lectures on quadratic forms, Tata Institute of Fundamental Research Lectures on Mathematics, No. 7, Tata Institute of Fundamental Research, Bombay, 1967. Notes by K. G. Ramanathan. MR 0271028
- Carl Ludwig Siegel, Über die Zetafunktionen indefiniter quadratischer Formen, Math. Z. 43 (1938), no. 1, 682–708 (German). MR 1545742, DOI 10.1007/BF01181113
Additional Information
- © Copyright 1985 American Mathematical Society
- 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