571p02 - f Z [ x ] is of degree d modulo p , then f ( x ) 0...

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MATH 571 ANALYTIC NUMBER THEORY I, SPRING 2011, PROBLEMS 2 Due 25th January 1. Prove that if f is a multiplicative function, then so is g ,deFnedby g ( n )= ° m | n f ( m ) . 2. Prove that ° m | n μ ( m )= ± 1 n =1 , 0 n> 1 . 3. Prove that φ ( n )= n ² m | n μ ( m ) m . 4. Prove that ² m | n φ ( m )= n . 5. Prove that if p is a prime number, d is a non-negative integer and
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Unformatted text preview: f Z [ x ] is of degree d modulo p , then f ( x ) 0 (mod p ) has at most d solutions. 6. Suppose that m N . Prove that if there is at least one primitive root modulo m , then there are exactly ( ( m ))....
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