Stadies on Andytic Priepeitied of autoinoiptic L-functions

首先，我们发展了模关系理论，并从模关系的角度统一处理了泛函方程、近似泛函方程和渐近展开式。从这个观点出发，我们给出了Katsurada和Matsumoto在Dirichlet l -函数和Hurwitz -函数上的渐近展开公式的模理论证明。其次，在Siegel模形式的Ikeda升降机上，我们给出了Kohnen关于Ikeda升降机象空间猜想的一部分。我们证明了标准l函数的均方的一个相当尖锐的渐近公式，它附属于模形式，即Ikeda提升。特别地，应用一个大值引理，我们在某些情况下几乎确定了均方的真实阶数。对于多重zeta函数的理论，我们开展了多重zeta值的双重洗牌关系的研究，并阐明了它们的算法结构。我们还发现了多重ζ函数之间的泛函关系可以用基础李代数的Weyl群的对称性来解释。此外，我们还发现多重ζ函数族的递归结构可以用Dynkin图来描述。

First, we have developed the theory of modular relations and have given a unified treatment of functional equations, approximate functional equations and asymptotic expansions in view of modular relations. From this viewpoint, we gave a modular-theoretic proof of asymptotic expansion formulas of Katsurada and Matsumoto on Dirichlet L-functions and Hurwitz zeta-functions. Secondly, on Ikeda liftings of Siegel modular forms, we have shown-a part of Kohnen's conjecture on the image space of Ikeda liftings. We have proved rather sharp asymptotic formula for the mean square of standard L-functions attached to modular forms which are Ikeda lifts. In particular, applying a large value lemma, we almost determined the true order of the mean square in some cases. As for the theory of multiple zeta-functions, we have developed the study of double shuffle relations of multiple zeta values, and have clarified their arithmetic structure. Also we discovered that the functional relations among multiple zeta-functions can be explained by the symmetry of Weyl groups of underlying Lie algebras. Moreover we have found that the recursive structure of the family of multiple zeta-functions can be described in terms of Dynkin diagrams.