Number-Theoretic Indentities・Asymptotic formulas,and Special Functions

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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.