571p07 - Math 571 Analytic Number Theory I, Spring 2011,...

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Math 571 Analytic Number Theory I, Spring 2011, Problems 7 Due Tuesday 1st March 1. (Vaughan 1973). Suppose that for each prime p ,0 b ( p ) <p and deFne L ( Q )= ° q Q μ ( q ) 2 ± p | q b ( p ) p b ( p ) . (a) Prove that L ( Q )= ° m ° r, Ω( r )= m s ( r ) Q ± p h ° r ² b ( p ) p ³ h where s ( r ) is the squarefree kernel of r and Ω( m ) is the total number of prime factors of m . (b) Prove that ° r, Ω( r )= m s ( r ) Q ± p h ° r ² b ( p ) p ³ h 1 m ! ° p Q 1 /m b ( p ) p m . (c) Suppose that ´ p X b ( p ) p F ( X ) > 0 for every X> 0. Prove that L ( Q ) max m exp µ m log m 1 F ( Q 1 /m ) · ¸ . 2. (Known as “Rankin’s Trick” but Rankin said that it was shown to him by Ingham!). Suppose that Q 1, and P N . Let L ( Q,P )= ° q Q,q | P μ ( q ) 2 ± p | q b ( p ) p b ( p ) . (a) Prove that, with the obvious abuse of notation, L ( ,P )= ¹ p | P p p b ( p ) . (b) Suppose that θ 0. Prove that L ( ,P ) L ( Q,P ) Q θ ° q | P q θ μ ( q ) 2 ± p | q b ( p ) p b ( p ) and L ( Q,P ) L ( ,P ) 1 Q θ ± p | P ² 1+( p θ 1) b ( p ) p ³ .
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This note was uploaded on 01/23/2012 for the course MATH 571 taught by Professor Vaughan,r during the Spring '08 term at Penn State.

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