Exponential Sums and the Riemann Zeta Function III

There is a serious error on p. 457. The sum over b cannot be taken inside the integral, and the factor H/U has to be removed from equation (4.13). The second result (1.7) of Theorem 1 (on power sums) is not valid. The first result (1.6) of Theorem 1, and Theorems 2 and 3 (on exponential sums) still hold. However, we must first replace the exponents5 4 ,1 4in (5.4) of Lemma 5.1 by4 3 ,1 3respectively. This improvement comes from the observation that in Lemma 5.1, if | k 2 – K j |δ 1 3H for all possible numberings, then the ten integers h 1 , …, h 5 , k 1 , …, k 5 must lie in an interval of length O (δ 2 9H ). There are also some typing errors: p. 453: the exponent of x should be − 1 2in the last term of f ( x ); p. 457: 12 should be 24 in | G ( 3 )|; p. 460: in Lemma 5.2, N 8 should have parameters (δ, b δ T ); p. 461: modulus signs are needed on the left of equation (5.11); p. 462: the first N 10 ′ in the fourth line should be N 10 .