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Abstract

Let α i (1≤is) be a set of nonzero integers. The form
$$A(\lambda ) = \sum\limits_{{i = 1}}^{s} {{{\alpha }_{i}}\lambda _{i}^{k}}$$
is called an additive form, and the equation
$$A(\lambda ) = 0$$
(8.1)
its corresponding additive equation. Let R s (0) be the number of solutions of (8.1) subject to the condition: λ i P(T), 1 ≤ Is. Then R s (0) can be expressed as an integral over U n ; see §5.1. The corresponding singular series is
$$\mathfrak{S}(0) = \sum\limits_{\gamma } {G(\gamma ) = \sum\limits_{a} {H(\mathfrak{a}),} }$$
where
$$G(\gamma ) = \prod\limits_{{i = 1}}^{s} {{{G}_{i}}(\gamma ),}$$
$$\begin{array}{*{20}{c}} {{{G}_{i}}(\gamma ) = N{{{({{\mathfrak{a}}_{i}})}}^{{ - 1}}}\sum\limits_{{\lambda ({{\mathfrak{a}}_{i}})}} {E({{\alpha }_{i}}{{\lambda }^{k}}\gamma ),} } & {\gamma {{\alpha }_{i}}\delta \to {{\mathfrak{a}}_{i}},} & {1 \leqslant i \leqslant s,} \\ \end{array}$$
and where
$$H(\mathfrak{a}) = {{\mathop{\sum }\limits_{\gamma } }^{ \star }}G(\gamma )$$
in which γ runs over a reduced system of (αδ)-1 mod δ-1.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Wang Yuan
    • 1
  1. 1.Institute of MathematicsAcademia SinicaBeijingChina

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