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Exponential Sums and the Riemann Zeta Function III
Author(s) -
Huxley M. N.,
Kolesnik G.
Publication year - 1993
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-66.2.302-s
Subject(s) - mathematics , lemma (botany) , exponent , exponential function , riemann hypothesis , combinatorics , exponential sum , riemann zeta function , function (biology) , discrete mathematics , pure mathematics , mathematical analysis , ecology , philosophy , linguistics , poaceae , evolutionary biology , biology
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 .