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# 571p12 - N k q h = ° χ χ k = χ χ h 3 Suppose that k |...

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Math 571 Analytic Number Theory I, Spring 2011, Problems 12 Due Tuesday 12th April Given positive integers k and q define G k ( q, a ) = q n =1 e an k q and let N k ( q, h ) denote the number of solutions of the congruence x k h (mod q ). p denotes a prime number. 1. (a) Prove that G k ( p, a ) = p h =1 N k ( p, h ) e ah p (b) Let l = ( k, p 1). Prove that N k ( p, h ) = N l ( p, h ) for all h , and hence that G k ( p, a ) = G l ( p, a ). (c) Prove that p 1 a =1 | G k ( p, a ) | 2 = p p h =1 N k ( p, h ) 2 p 2 . (d) Suppose that k | ( p 1). Prove that there are ( p 1) /k non-zero residues h (mod p ) for which N k ( p, h ) = k , that N k ( p, 0) = 1, and that N k ( p, h ) = 0 for all other residue classes (mod p ). Prove that the right hand side of (c) is p ( p 1)( k 1) and hence that in general it is p ( p 1)(( k, p 1) 1). (e) Suppose that p a , p c , and that b ac k (mod p ). Prove that G k ( p, a ) = G k ( p, b ). (f) Suppose that k | ( p 1). Prove that if p a then | G k ( p, a ) | < k p , and hence that in general | G k ( p, a ) | < ( k, p 1) p . 2. Suppose that ( h, q ) = 1. (a) Prove that 1 ϕ ( q ) χ χ ( x ) k χ ( h ) = 1 if x k h (mod q ) , 0 otherwise. (b) Prove that
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Unformatted text preview: N k ( q, h ) = ° χ χ k = χ χ ( h ) . 3. Suppose that k | ( p − 1) and let χ be a character modulo p oF order k , say χ ( n ) = e ((ind n ) /k ). (a) Prove that For all h , N k ( p, h ) = 1 + k − 1 ° j =1 χ ( h ) j . (b) Prove that iF p ° a , then G k ( p, a ) = k − 1 ° j =1 χ ( a ) j τ ( χ j ) . (c) Prove that iF p ° a , then | G k ( p, a ) | ≤ ( k − 1) √ p and hence in general | G k ( p, a ) | ≤ (( k, p − 1) − 1) √ p . 4. Let M s ( q, n ) denote the number oF solutions in x 1 , . . . , x s oF the congruence x k 1 + · · · + x k s ≡ n (mod q ). (a) Prove that M s ( q, n ) = 1 q q ° a =1 G k ( q, a ) s e ( − an/q ). (b) Prove that | M s ( p, n ) − p s − 1 | ≤ (( k, p − 1) − 1) s p s/ 2 ....
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