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By Carlos J. Moreno

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Additional resources for Advanced Analytic Number Theory, Part I: Ramification Theoretic Methods

Example text

2 W A X l ) A \ /'i±£>\dx «i / ^(1-S- 1 )- 1 a i d a lim . |x|-2Xi(o:)^. / As for t h e s e c o n d i n t e g r a l , o n e h a s - ( 1 - 9 " " 1 ) " " 1 9 X i M l im / A -+™J\x\=i,\ai\=q = (l-q-1)-2Xi(zu)lim |1 + x | - 2 (<^AXi)(l + x)t/>(ai)dxai A J\x\=\l + x\=l + x\-2(AXi)(l+x)dx f \l + -+°°J\x\ = l (1 - AXi){l = (1 - ^ 1 ) - 2 X i ( ^ ) [ l - Zq-1 + dx Hm + x)dx] \x\'2(4>AXi){x)dx\. 3). T h i s c o m p l e t e s t h e e v a l u a t i o n i n t h i s case.

11) holds. Piece 2: \p + r\ — 1, |p + r — 1 | < 1. 9) yields \r\ = |pH-r — 1 | 2 < 1, and thus \p + r - 1| = \p - 1|. 11) holds. Piece 3: \p + r\ > 1. 10) imply that \p+r\ = \r\ = qh-fl\p\. In particular |p| < |r|, so f2 > / i . 6) restricted to this domain: r — i > r — p. rj)w- 2 r(p - 1) dpdr where the integration is over \r\ = \r-p\ = qh~h\p\ > 1. Send p i—• p ( l + v), \v\ < 1, \v\ < |1 — p _ 1 | . Then the x i integral is independent of v (resp. oscillates) if \v\ < q~fl\p — 1| (resp. \v\ > q~fl\p — 1|), while th e \2 integral oscillates (resp.

10) imply that \p+r\ = \r\ = qh-fl\p\. In particular |p| < |r|, so f2 > / i . 6) restricted to this domain: r — i > r — p. rj)w- 2 r(p - 1) dpdr where the integration is over \r\ = \r-p\ = qh~h\p\ > 1. Send p i—• p ( l + v), \v\ < 1, \v\ < |1 — p _ 1 | . Then the x i integral is independent of v (resp. oscillates) if \v\ < q~fl\p — 1| (resp. \v\ > q~fl\p — 1|), while th e \2 integral oscillates (resp. 1 \v\ > 1 (resp. q*1 \v\ < 1). 11) holds. 6. e. th e integral restricted to this subdomain vanishes unless |p—1| = 1.

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