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571p08 - R be R real numbers such that ° x r − x s °...

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Math 571 Analytic Number Theory I, Spring 2011, Problems 8 Due Tuesday 15th March 1. (Gallagher 1967.) (a) Suppose that α> 0 and f ° is continuous on [ α,α ]. By considering ° 0 α f ( β ) and ° α 0 f ( β ) separately and integrating by parts, or otherwise, prove that 2 αf (0) = ± α α f ( β ) + ± 0 α f ° ( β )( β + α ) + ± α 0 f ° ( β )( β α ) and that | f (0) |≤ 1 2 α ± α α | f ( β ) | + 1 2 ± α α | f ° ( β ) | dβ. (b) Let K Z , K 0, 0 1 2 and U ( x )= K ² k = K b k e ( αk ) . Prove that | U ( x ) | 2 1 δ ± x + δ/ 2 x δ/ 2 | U ( β ) | 2 + ± x + δ/ 2 x δ/ 2 | U ( β ) U ° ( β ) | dβ. (c) Prove that ± 1 0 | U ( β ) | 2 = K ² k = K | b k | 2 and ± 1 0 | U ° ( β ) | 2 4 π 2 K 2 K ² k = K | b k | 2 . (d) Let x 1 ,x 2 ,...,x
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Unformatted text preview: R be R real numbers such that ° x r − x s ° ≥ δ ( r ± = s ). Prove that R ² r =1 | U ( x r ) | 2 ≤ ³ 2 πK + 1 δ ´ K ² k = − K | b k | 2 . (e) Let T ( x ) = M + N ² n = M +1 a n e ( xn ) . Prove that R ² r =1 | T ( x r ) | 2 ≤ ³ πN + 1 δ ´ M + N ² n = M +1 | a n | 2 Hint. Let K = µ N 2 ¶ , b k = a k + M +1+ K ( − K ≤ k ≤ N − K − 1), b K = 0 when N is even....
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