571p05 - P and P = ´ p ≤ z p(b Let Λ d be real with Λ...

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MATH 571 ANALYTIC NUMBER THEORY I, SPRING 2011, PROBLEMS 5 Due 15th February ° n z μ ( n ) 2 ϕ ( n ) ± ° d | q μ ( d ) 2 ϕ ( d ) ²± ° m z ( m,q )=1 μ ( m ) 2 ϕ ( m ) ² . (b) Deduce that ° n z ( n,q )=1 μ ( n ) 2 ϕ ( n ) ϕ ( q ) q ° n z μ ( n ) 2 ϕ ( n ) . λ + d be an upper bound sifting function such that λ + d = 0 for all d>z . Show that for any q , 0 ϕ ( q ) q ° d ( d,q )=1 λ + d d ° d λ + d d . (Hint: Multiply both sides by P/ϕ ( P )= ³ 1 /m where m runs over all integers composed of the primes dividing
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Unformatted text preview: P , and P = ´ p ≤ z p .) (b) Let Λ d be real with Λ d = 0 for d > z . Show that for any q , ≤ ϕ ( q ) q ° d, e ( de,q )=1 Λ d Λ e [ d, e ] ≤ ° d, e Λ d Λ e [ d, e ] . (c) Let λ − d be a lower bound sifting function such that λ − d = 0 for d > z . Show that if each prime divisor p of q satisFes p ≤ z , then ϕ ( q ) q ° d ( d,q )=1 λ − d d ≥ ° d λ − d d ....
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