Unformatted text preview: s . Exercise 3. Show that the equation x 1 + x 2 + Â·Â·Â· + x j = n has Â± n1 j1 Â¶ solutions in positive integers. Note: solutions diï¬€er if they diï¬€er in at least one index, i.e. (1 , 1 , 2) diï¬€ers from (2 , 1 , 1). Exercise 4. Show that there are 2 n1 ways of writing n as a sum of strictly positive integers where order of summands is taken into account (i.e., 1 + 2 + 1 is diï¬€erent from 2 + 1 + 1 and both are diï¬€erent from 1 + 1 + 2). Exercise 5. Prove that the number of positive fractions â‰¤ 1 that have denominator â‰¤ n (when reduced to lowest terms) is Ï† (1) + Ï† (2) + Ï† (3) + Â·Â·Â· + Ï† ( n ) . Exercise 6. Assume mÏ† ( m ) = nÏ† ( n ). Show m = n . Exercise 7. Let N be a positive integer and let f and F be arithmetic functions. Show that the following are equivalent: 1. F ( n ) = âˆ‘ b N/n c k =1 f ( kn ) for all n . 2. f ( n ) = âˆ‘ b N/n c k =1 Î¼ ( k ) F ( nk ) for all n . (Here Î¼ is the MÂ¨obius function.)...
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This note was uploaded on 04/05/2010 for the course CS 150 taught by Professor Humphreys,g during the Spring '08 term at UVA.
 Spring '08
 Humphreys,G

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