Publicacions Matemàtiques

Bandlimited Approximations and Estimates for the Riemann Zeta-Function

Emanuel Carneiro, Andrés Chirre, and Micah B. Milinovich

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Abstract

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.

Article information

Source
Publ. Mat., Volume 63, Number 2 (2019), 601-661.

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

Permanent link to this document
https://projecteuclid.org/euclid.pm/1561687235

Digital Object Identifier
doi:10.5565/PUBLMAT6321906

Mathematical Reviews number (MathSciNet)
MR3980935

Zentralblatt MATH identifier
07094864

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 41A30: Approximation by other special function classes

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

Citation

Carneiro, Emanuel; Chirre, Andrés; Milinovich, Micah B. Bandlimited Approximations and Estimates for the Riemann Zeta-Function. Publ. Mat. 63 (2019), no. 2, 601--661. doi:10.5565/PUBLMAT6321906. https://projecteuclid.org/euclid.pm/1561687235


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