ZETA FUNCTIONS OF GROUPS AND RINGS

By ANTHONY WILLIAMS

CONTENTS INTRODUCTION A BRIEF HISTORY OF ZETA FUNCTIONS EULER, RIEMANN DIRICHLET DEDEKIND ARTIN, WEIL BIRCH, SWINNERTON-DYER ZETA FUNCTIONS OF GROUPS ZETA FUNCTIONS OF ALGEBRAIC GROUPS ZETA FUNCTIONS OF RINGS LOCAL FUNCTIONAL EQUATIONS UNIFORMITY ANALYTIC PROPERTIES P-ADIC INTEGRALS NATURAL BOUNDARIES OF EULER PRODUCTS NILPOTENT GROUPS: EXPLICIT EXAMPLES CALCULATING ZETA FUNCTIONS OF GROUPS CALCULATING ZETA FUNCTIONS OF LIE RINGS CONSTRUCTING THE CONE INTEGRAL RESOLUTION EVALUATING MONOMIAL INTEGRALS SUMMING THE RATIONAL FUNCTIONS EXPLICIT EXAMPLES FREE ABELIAN LIE RINGS HEISENBERG LIE RING AND VARIANTS GRENHAM’S LIE RINGS FREE CLASS-2 NILPOTENT LIE RINGS THREE GENERATORS N GENERATORS THE ‘ELLIPTIC CURVE EXAMPLE’ OTHER CLASS TWO EXAMPLES 2 10 THE MAXIMAL CLASS LIE RING M3 AND VARIANTS 2 11 LIE RINGS WITH LARGE ABELIAN IDEALS 2 12 F3,2 2 13 THE MAXIMAL CLASS LIE RINGS M4 AND FIL4 2 14 NILPOTENT LIE ALGEBRAS OF DIMENSION ?6 2 15 NILPOTENT LIE ALGEBRAS OF DIMENSION 7 SOLUBLE LIE RINGS PROOF OF THEOREM 3 1 CHOOSING A BASIS FOR TRN(Z) DETERMINING THE CONDITIONS CONSTRUCTING THE ZETA FUNCTION TRANSFORMING THE CONDITIONS DEDUCING THE FUNCTIONAL EQUATION VARIATIONS QUOTIENTS OF TRN(Z) COUNTING ALL SUBRINGS ALGEBRAIC GROUPS NILPOTENT GROUPS AND LIE RINGS THE CONJECTURE SPECIAL CASES KNOWN TO HOLD A SPECIAL CASE OF THE CONJECTURE PROJECTIVISATION MANIPULATING THE CONE SUMS CONES AND SCHEMES QUASI-GOOD SETS QUASI-GOOD SETS: THE MONOMIAL CASE APPLICATIONS OF CONJECTURE 4 5 COUNTING SUBRINGS AND P-SUBRINGS COUNTING IDEALS AND P-IDEALS HEIGHTS, COCENTRAL BASES AND THE ?-MAP PROPERTY (†) LIE RINGS WITHOUT (†) NATURAL BOUNDARIES I: THEORY A NATURAL BOUNDARY FOR ?GSP6(S) NATURAL BOUNDARIES FOR EULER PRODUCTS PRACTICALITIES DISTINGUISHING TYPES I, II AND III AVOIDING THE RIEMANN HYPOTHESIS ALL LOCAL ZEROS ON OR TO THE LEFT OF ?(S) = ? USING RIEMANN ZEROS AVOIDING RATIONAL INDEPENDENCE OF RIEMANN ZEROS CONTINUATION WITH FINITELY MANY RIEMANN ZETA FUNCTIONS IN?NITE PRODUCTS OF RIEMANN ZETA FUNCTIONS NATURAL BOUNDARIES II: ALGEBRAIC GROUPS G = GO2L+1 OF TYPE BL G = GSP2L OF TYPE CL OR G = GO+ 2L OF TYPE DL G = GSP2L OF TYPE CL G = GO+ 2L OF TYPE DL NATURAL BOUNDARIES III: NILPOTENT GROUPS ZETA FUNCTIONS WITH MEROMORPHIC CONTINUATION ZETA FUNCTIONS WITH NATURAL BOUNDARIES TYPE I TYPE II TYPE III OTHER TYPES TYPES IIIA AND IIIB TYPES IV, V AND VI