extra-homework5

extra-homework5 - p n n log n ? 4. Let p n be as in problem...

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Extra Problems 3/3/09 1. Prove that ( m ) m 3 for m 2 Z , m ± 33 : 2. Let ± ( x ) be the number of integers 1 n x such that n is a square prove that lim x 1 + 1 ± ( x ) ( x ) = 0 : 3. Let p n be the n th prime (in pari gp p n = prime ( n ) ). Calculate p n n log( n ) for n = k (1000) ; k = 20 ; 40 ; 60 ; 80 ; 100 (you will have to increase prime limit). What would you guess about lim n !1
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Unformatted text preview: p n n log n ? 4. Let p n be as in problem 3. Use the fact that & ( p n ) = n to prove that there exist constants A; B > such that An log n & p n & Bn log n: (Hint: Show that there is a constant, C , such that p n & Cn 2 .) 1...
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