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Proof. The upper bound of (2.7) follows immediately fromsince0 1 ./ k ./ D X .k1jn " p j11 C . 1/p j1/ XIn particular, when D 1, (2.7) has the formFor (2.8), we have 1 ./ D X 1 21jnp jp 2 j11 C . 1/p j 2 #1 p k 1j1 C . 1/p j.minf1; g/ 2 .k 1/ 2 ;1 C . 1/p j D q j C p j minf1; g: (2.9)1jnX1jnp j 10 k .k 1/ 2 : (2.10)p j X1 C . 1/p jp j 1 2X1jnp j 11jnp j 1p j 1 C . 1/p jp j .1 p j / D 2 . 2/:Lemma 2.5. For r D 1 C x=,.r 1/20 m 1 .r/ . 2 3 /minf1; rg ; (2.11)and, if x D o./ as ! 1, then1p1 .r/ 2 .r/ D 1 Proof. We have1 3 2 C 2 32 3x C O x2 2 : (2.12) 1 .r/ 2 .r/ D r X p j pj2.1 C .r 1/p j / 21jnD r 2 1 2 2 3 3 4x C O 3 4x 2 ;which, together with the estimate r 1=2 D 1 x=.2/ C O x 2 = 2 , yields (2.12). Note that the factor. 3 2 C 2 3 /= 3 is of order 1 .Similarly, since m D C.r 1/ 2 , we have m 1 .r/ D .r 1/ P 2 1jn p2 j .1 p j/=.1C.r 1/p j /,from which (2.11) follows.9