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Exponential Sums and the Riemann Zeta Function III
Author(s) -
Huxley M. N.,
Kolesnik G.
Publication year - 1991
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/s3-62.3.449
Subject(s) - mathematics , riemann zeta function , exponential function , riemann hypothesis , square root , square (algebra) , function (biology) , riemann xi function , particular values of riemann zeta function , combinatorics , exponential sum , pure mathematics , arithmetic zeta function , mathematical analysis , prime zeta function , geometry , evolutionary biology , biology
Let F ( x ) be a real function with sufficiently many derivatives existing and satisfying certain non‐vanishing conditions for 1 ⩽ x ⩽ 2. We improve the estimates for the exponential sum S = ∑ m e ( T F ( m M ) )especially when M is close to the square root of T . For the Riemann zeta function, ξ ( 1 2 + i T ) ≪ T 17 / 108 + ∈the partial sums of length about T ½ are no longer critical. We use the ‘discrete Hardy‐Littlewood method’ of Bombieri and Iwaniec. The improvement lies in counting the number of approximate equalities among certain power‐sums in the square roots of integers.