Exponential Sums and the Riemann Zeta Function IV | Zendy

Huxley M. N. | Zendy

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Exponential Sums and the Riemann Zeta Function IV

Author(s) -

Huxley M. N.

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/s3-66.1.1

Subject(s) - mathematics , riemann zeta function , exponential function , riemann hypothesis , function (biology) , square root , particular values of riemann zeta function , pure mathematics , exponential decay , exponential growth , riemann xi function , exponential sum , arithmetic zeta function , mathematical analysis , combinatorics , prime zeta function , geometry , quantum mechanics , physics , 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 sumS = ∑ m e ( T F ( m M ) )especially when M is close to the square root of T . For the Riemann zeta function, ζ(½+ iT )≪ T 89/570+ɛ . We use the ‘discrete Hardy‐Littlewood method’ of Bombieri and Iwaniec. The improvement lies in comparing rational approximations to the derivatives of F at nearby integers m .

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