Number-Theoretic Indentities・Asymptotic formulas,and Special Functions

在这三年中获得的最大成果是发现了贯穿Zeta函数理论的模关系--模关系原理的路线，这是我们在1999年12月发现的，2001年，我们将这一原理应用于Hecke型函数方程的特殊情况，即这里是一个简单的Gamma因子的情况，它包括了Zeta函数的许多重要的特殊情况，包括多重Hurwitz Zeta函数，Epstein Zeta函数，自同构L函数等。特别地，我们成功地得到了这些Zeta函数在有理变元和积分变元上的特殊值的Raveljan型快速收敛公式。我们是通过我们在第一篇论文中完成的关于用Hurwitz Zeta函数的导数表示具有Hurwitz Zeta函数系数的无穷级数的求和公式的推导得出这一发现的，特别是，我们的结果取代了最近出版的Srivastava和Choi的书“与Zeta和相关函数相关的级数”中的所有相应结果。与此沿着，我们继续对梅勒函数进行了不断的研究，并成功地证明了在积分变元中，Dedeking Zeta函数的特殊值的近似表达式，方法是将它们与克劳森函数相结合，这一点在其他文献中被忽略了。

THE GREATEST RESULT THAT IS OBTAINED DURING THESE THREE YEARS IS THE DISCOVERY OF THE ROUTE PENETRATING THE MODULAR RELATIONS-MODULAR RELATION PRINCIPLE-FOR THE THEORY OF ZETA-FUNCTIONS, WHICH WE FOUND IN DEC., 2000.IN 2001, WE APPLIED THIS PRINCIPLE TO THE SPECIAL CASE OF THE FUNCTIONAL EQUATIONS OF HECKE'S TYPE-I.E. THE CASE WHERE HERE IS A SIMPLE GAMMA FACTOR, WHICH INCLUDES MANY OF THE MOST IMPORTANT SPECIAL CASES OF ZETAFUNCTIONS INCLUDING MULTIPLE HURWITZ ZETA-FUNCTION, EPSTEIN ZETA-FUNCTION, AUTOMORPHIC L-FUNCTION, ETC.ESPECIALLY, WE WERE SUCCUSSFUL IN OBTAINING THE RAMANUJAN TYPE RAPIDLY CONVERGENT FORMULAS FOR SPECIAL VALUES OF THOSE ZETA-FUNCTIONS AT RATIONAL AS WELL AS INTEGRAL ARGUMENTS. WE CAME TO THIS DISCOVERY THROUGH OUR FORMER INVESTIGATIONS ON THE SUM FORMULA EXPRESSING INFINITE SERIES WITH HURWITZ ZETA-FUNCTION COEFFICENTS IN TERMS OF THE DERIVATIVES OF THE HURWITZ ZETA-FUNCTION, WHICH WE COMPLETED IN THE FIRST LISTED PAPER, IN PARTICULAR, OUT RESULTS SUPERSEDE ALL THE CORRESPONDING RESULTS IN THE RECENTLY PUBLISHED BOOK OF SRIVASTAVA AND CHOI "SERIES ASSOCIATED WITH THE ZETA AND RELATIED FUNCTIONS".ALONG WITH THIS WE CONTINUED OUR NEVER-STOPPING RESEARCH ON MAILLET DETERMINATS, AND SUCCEEDED IN PROVING THE DETERMINATAL EXPRESSIONS FOR THE SPECIAL VALUES OF THE DEDEKING ZETA-FUNCTION AT INTEGRAL ARGUMENTS, BY COMBINING THOSE WITH CLAUSEN FUNCTIONS, A POINT WHICH IS OVERLOOKED IN OTHER LITERATURE.