Download e-book for kindle: Analytic Theory of Continued Fractions III: Proceedings of a by Lisa Jacobsen

By Lisa Jacobsen

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U>1 , and let Next, set Then . 1 with N = 1 , and defining D,(x) 1dt(n) n a we obtain W e have, since Ibt(n) I 17 (log 2n) 3 n=l << log t-2 x . M. RE KONINCK AND A. IVI6 On the other hand, =logt-10 c . nsx bt(n) (1 t o[*]] bt ( n ) 17 t o(logt-2x) . =logt-1 x . 1. 1. Let S denote the set of all arithmetical func- tions satisfying the following two conditions: * 1) f ( n ) 2) nsx 0 => 1=0(---). f(n) L 1 , for each positive integer n . 2. 1 with D(t) E catl C0,ll . 3. 2; then for t E D(t) (0,ll be the cor- , let D ( t ) (i-1) B .

M. RE KONINCK AND A. IVI6 On the other hand, =logt-10 c . nsx bt(n) (1 t o[*]] bt ( n ) 17 t o(logt-2x) . =logt-1 x . 1. 1. Let S denote the set of all arithmetical func- tions satisfying the following two conditions: * 1) f ( n ) 2) nsx 0 => 1=0(---). f(n) L 1 , for each positive integer n . 2. 1 with D(t) E catl C0,ll . 3. 2; then for t E D(t) (0,ll be the cor- , let D ( t ) (i-1) B . ( t ) = (-) t Z and A i ( t ) = (-l)i-l B i ( t ) write Ai for A i ( l ) , for 1,2,.. ,a+2 i . We shall occasionally 84.

Proof. 6. 86. A generalization of the main theorem. 4. 1, with ~ ( t= )g(l,t)/r(t) E . c~+~co,~I From this definition, we observe that if (g,f) uniformly for It I s 1 . 3. now state the following theorem. 8. Let @,f) E S; .

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