# Analytic Arithmetic in Algebraic Number Fields by Baruch Z. Moroz

By Baruch Z. Moroz

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Extra info for Analytic Arithmetic in Algebraic Number Fields

Example text

200, Theorem 4): [Ck(s)l in the strip i l+sl ( / ~ ~ 3[T~ -q < Res < 1+q. ll+sl 2~ We write )nl2)1+n-Re S~(l+~)n (3o) 39 L(s,x) (with (17) the g(x') has = O) already identical completes Remark and been one, I. apply (30) proved we may the p r o o f to the apply of first factor for r e p r e s e n t a t i o n s in which it to the s e c o n d of p r o p o s i t i o n In n o t a t i o n s (31) = ~k(s)g(X)L(s,x') factor (31). Since do n o t c o n t a i n in (31). This 2. 39), we have m j=~+1 Lemma 5.

89], by assumption. 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.

43) ejxj an equation: m X{P) = Ip[