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A note on the mean square of |ζ(½ + it )|
Author(s) -
Watt N.
Publication year - 2010
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdq024
Subject(s) - mathematics , exponent , upper and lower bounds , combinatorics , bounded function , square (algebra) , constant (computer programming) , integer (computer science) , function (biology) , exponential function , divisor (algebraic geometry) , discrete mathematics , mathematical analysis , geometry , philosophy , linguistics , evolutionary biology , computer science , biology , programming language
Using the method of Bombieri and Iwaniec, new upper bounds are obtained for the absolute value of exponential sumsS = ∑ H / 2 < h ⩽ H∑ M / 2 < m ⩽ M exp ( 2 π i T F ( m + h M ) − 2 π i T F ( m − h 1 M ) )with parameters T and M large, T 1/3 < M < T 2/3 and 1 ⩽ H < MT −17/57 , and with certain non‐vanishing conditions imposed on the derivatives of the function F : [1/3, 3]↦ℝ. For E ( T ), the error term in the asymptotic formula for the mean square of |ζ(½ + it )|, one consequently obtains a new bound E ( T ) ≪ T θ log ϕ ( T + 2), with a constant ϕ < 4, and with the same ‘main exponent’, θ = 131/416, as occurs in the sharpest upper bounds yet found for the error terms of the circle and divisor problems. The proofs utilize recent advances by Huxley, applying to the Bombieri–Iwaniec method in general, and independent progress of the author on a question specific to the sum S . Conditional improvements are given, subject to Huxley's ‘Hypothesis H ( κ, λ )’ concerning the number of integer points near a given bounded curve.