Publicacions Matemàtiques

Bandlimited Approximations and Estimates for the Riemann Zeta-Function

Emanuel Carneiro, Andrés Chirre, and Micah B. Milinovich

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation