Open Access
2019 Bandlimited Approximations and Estimates for the Riemann Zeta-Function
Emanuel Carneiro, Andrés Chirre, Micah B. Milinovich
Publ. Mat. 63(2): 601-661 (2019). DOI: 10.5565/PUBLMAT6321906


In this paper we provide explicit upper and lower bounds for the argument of the Riemann zeta-function and its antiderivatives in the critical strip under the assumption of the Riemann hypothesis. This extends the previously known bounds for these quantities on the critical line (and sharpens the error terms in such estimates). Our tools come not only from number theory, but also from Fourier analysis and approximation theory. An important element in our strategy is the ability to solve a Fourier optimization problem with constraints, namely, the problem of majorizing certain real-valued even functions by bandlimited functions, optimizing the $L^1(\mathbb{R})$-error. Deriving explicit formulae for the Fourier transforms of such optimal approximations plays a crucial role in our approach.


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Emanuel Carneiro. Andrés Chirre. Micah B. Milinovich. "Bandlimited Approximations and Estimates for the Riemann Zeta-Function." Publ. Mat. 63 (2) 601 - 661, 2019.


Received: 24 October 2017; Published: 2019
First available in Project Euclid: 28 June 2019

zbMATH: 07094864
MathSciNet: MR3980935
Digital Object Identifier: 10.5565/PUBLMAT6321906

Primary: 11M06 , 11M26 , 41A30

Keywords: argument , Beurling-Selberg extremal problem , critical strip , exponential type , extremal functions , Gaussian subordination , Riemann hypothesis , Riemann zeta-function

Rights: Copyright © 2019 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.63 • No. 2 • 2019
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