2. QUANTITATIV E RELATION S 9 PROOF. I t suffice s t o show that D(N', a , (3) \N{(3 - a). Pro m the Theore m we see tha t D{N 'a' ® - YTi + 2 ( ^ T I + 0 ~ a) £ I £ e{kUl k=\'n=l which b y hypothesi s i s GfTTn*TT+"-")w- Since -^ (/? - a)/4 , thi s give s the desire d bound . Let E K {N) denot e the right hand side of (24), and put E(N) = min* E K (N). Thus E'(iV ) i s th e bes t uppe r boun d fo r D*(N) tha t ca n b e derive d fro m th e Erdos-Turan inequality . A n alternativ e uppe r boun d fo r D(N) ha s bee n give n by LeVequ e [7] , namely / fi ° ° \ 1 / 3 (26) D(N)(±Nj2\UN(k)f/k 2 ) . Let -L(iV) denote the right hand side above. W e now show that E(N) C L(N), so tha t th e boun d provide d b y th e Erdos-Tura n inequalit y i s alway s th e bette r of the two , apar t fro m a constant factor . THEOREM 2 . Let E(N) and L(N) be the upper bounds for the discrepancy D(N) provided by the Erdos-Turan and LeVeque inequalities. Then (27) D(N) E(N) « L(N) « N 1/3 D(N)2/3. PROOF. Th e first inequalit y i s the Erdos-Turan inequalit y (24) . T o prove th e second inequality , tak e K = [2N/E{N)\. The n ^ \E{N), s o tha t E{N)^Y^\\UN{k)\. By Cauchy' s inequalit y thi s i s K . v i / 2 **l/2(£^(*)ia) That is , (^) (Epifew f 00 1 \V 2 ^ ( 7 V ) 3 / 2 « ( 7 V ^ ^ | C / N ( f c ) | 2 which give s the state d result .

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