By Baruch Z. Moroz

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**Extra resources for Analytic Arithmetic in Algebraic Number Fields**

**Sample text**

We h a v e f(- +it) = F (+it)r(~-it), from the p r o p e r t i e s t • ~+. , 35 I r(~) f(- +it) < and we deduce (10) from Suppose that L(s,x) 7 r()4 t E~+ I I a = -4' b = 2' A = 4, ~ = O. 26) with 1 for I < j < m' -I for m' + I <_ j _< m ej = Let m' m' $1 = 9=IU S I (kj), For p 6 S1(kj) tively, where Proposition we write X. I. IL~ (1-s,X) L~(s,X) Let definitions and tp is a grossencharacter for (13) n = j=m'+1~ [kj:~]. 26), and tp(Xj) respec- I < j < m. I [ap I+itpl 2) ~-Re s I <12n( H [1+s+itpl ~~ ll+s+ 2 ' PES I PES 2 (14) - ~1 _< Re s < ~I .

Ch. m a y be XIII). 26); write m 2 Z n. j=1 3 = satisfies (11), then 1-s g(x) ]~ (24n(x)£ n(x) e n(x) e) [L(s,x) (I-~) = 0a (1+Itl) (a(x)b(x)) I Re s > z+¢ for In p a r t i c u l a r , Remark I. slightly i) p ii) , e > O, (12) follows Alternatively stronger is of A W L(s,x) ~ O Definition I. ~ E {+I} one estimate type, for from can from , t:= Im s. the G e n e r a l i s e d deduce the (L 12) Riemann an a n a l o g o u s to Hypothesis. (12) b u t a two c o n d i t i o n s : and 1 Re s > 2" If r e l a t i o n (12) holds, we say that p and suppose is of L i n d e l ~ f tYPe.

14) = m e. n a(xj) ] j=1 (34) that the f u n c t i o n (35) A(s,x): = L ( s , x ) L ~ ( s , X ) equation ~-s A(I-s,X); A(s, X) = W(X) a(x) Definition type, I. Let p E R(k) if the f u n c t i o n Corollar[ 3. t h a t each p R(k) Proof. The c o n j e c t u r e suppose that a n d let p 6 R(k) X = tr (31) is h o l o m o r p h i c The A r t i n - W e i l in and conjecture is of A W clearly and [W(X) I = I. p. We say in C\{1 }. that is e q u i v a l e n t p (36) is of A W to the s t a t e m e n t type.