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2.5 The usual LLT for S nWith the above estimates available, we can now prove four LLTs, starting from the one in the usual form.Proposition 2.7. If ! 1 as n ! 1; thenP.S n D m/ D e x2 =2p2 1 C 3 2 C 2 36 3 .x 3 3x/ C Ouniformly for m D C x with jxj 1=3 . x 6 C 1 2: (2.16)Proof. By Cauchy’s (or Fourier’s) integral formulaP.S n .r/ D m/ D 1 Ze mit F.rI e it /dt C 1 Ze mit F.rI e it /dt2 jtjt 02 t 0