571p04 - x 0, y 1, Q ( x + y ) Q ( x ) y (2) + O y 2 / 3 ....

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MATH 571 ANALYTIC NUMBER THEORY I, SPRING 2011, PROBLEMS 4 Due 8th February 1. (H.-E. Richert, unpublished) (a) Show that ° x<n x + y ± ° d 2 | n Λ d ² 2 = y ° d, e Λ d Λ e [ d, e ] 2 + O ³ ± ° d | Λ d | ² 2 ´ . (b) Let f ( n )= n 2 µ p | n 1 p 2 · . Show that ¸ d | n f ( d )= n 2 . (c) For 1 d z let Λ d be real numbers such that Λ 1 = 1. Show that the minimum of ° d, e Λ d Λ e [ d, e ] 2 is 1 /L where L = ¸ n z μ ( n ) 2 /f ( n ). Show also that Λ d ° 1 for the extremal Λ d . (d) Show that ζ (2) 1 /z L ζ (2). (e) Let Q ( x ) denote the number of squarefree numbers not exceeding x . Show that for
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Unformatted text preview: x 0, y 1, Q ( x + y ) Q ( x ) y (2) + O y 2 / 3 . 2. (N. G. de Bruijn, unpublished; cf van Lint &amp; Richert (1964)) Let M be an arbitrary set of natural numbers, and let s ( n ) denote the largest squarefree divisor of n . Show that n x n M ( n ) 2 ( n ) n x s ( n ) M 1 n n x ( n ) 2 ( n ) n x 1 n 1 ....
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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 Pennsylvania State University, University Park.

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