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Exponential Sums and the Riemann Zeta Function
Author(s) -
Huxley M. N.,
Watt N.
Publication year - 1988
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/s3-57.1.1
Subject(s) - mathematics , riemann zeta function , exponential function , riemann hypothesis , function (biology) , gauss , poisson distribution , rational function , pure mathematics , arithmetic zeta function , number theory , rational point , interval (graph theory) , combinatorics , mathematical analysis , discrete mathematics , statistics , algebraic number , physics , quantum mechanics , evolutionary biology , biology
Let F ( x ) be a real function with sufficiently many derivatives existing and satisfying certain non‐vanishing conditions in the interval [1,2]. Then∑ m = M 2 M - 1e ( T F ( m M ) ) = O ( M 1 / 2T 9 / 56log T )as M, T → ∞ with T ⩾ M . Such sums often occur in the study of the zeta function. This result extends recent work of Bombieri and Iwaniec, whose ‘discrete Hardy–Littlewood method’ uses rational approximations to the derivatives of F ( x ) at rational points, Gauss sums, Poisson summation, and the large sieve.