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By Baruch Z. Moroz

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Extra resources for Analytic Arithmetic in Algebraic Number Fields

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Re u I _< ~. and (12) 2q 6 ZZ, while in (3) and q • {O,1}. I I a = - ~, b = ~ If ( + i t ) Therefore 4 that F(s+1) Let Re u whenever I F ( ' q + u ) - l F ' c {/ + l - 2u ) 1 2 In for hold: ,q+u. -1 1 )F( 2 J I < (~ll+ul) 2 u b-a case of T h e o r e m The f o l l o w i n g . ~ with then if(u) I < (AI~+ul~) Proof. t E]R whenever (12) and e s t i m a t e (10) with and the f u n c t i o n a l = sF(s), I q ~ ~ follow equation s • ~. We h a v e f(- +it) = F (+it)r(~-it), from the p r o p e r t i e s t • ~+.

By P(X,t) = 0 is a p o l y n o m i a l when constant g(X) = O, may d e p e n d on of degree g(x)-I x > 2; ~ > O. e and Here when the nd(x). (3), oo L(S,X) with an e f f e c t i v e l y Therefore p the line This Z n=1 nen-Sc1 (e,nd(x)), computable (in terms of e > 0 (36) nd(x) and c) C I > O. ~i Since << l+q+iT L(w,x) f %+D-iT is of A W type, Re s = O procedure 1+n I+c x + O( ~---~TrTT-C_ ) + Oe( x" one can m o v e passing leads w ~x- d w the r e s i d u e to the e s t i m a t e (log x) -I, T = x 2/N+2 IL(s,x) I < I the c o n t o u r and r e p l a c e s T of i n t e g r a t i o n of m u l t i p l i c i t y g(x) (35) as s o o n as one takes (17) by (I+q-I)nd(X)B(X) log X ) at to s = O.

6 R(k), L(s,xj) X = tr ~ / O for and suppose I Re s > ~ , that L(s,x) I < j < m.

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