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arXiv · 2403.17803

On Littlewood's estimate for the modulus of the zeta function on the critical line

Emanuel Carneiro·Micah B. Milinovich
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Abstract

Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. Our proof uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and give a slight refinement of the sharpest known upper bound (due to Chandee and Soundararajan) for the modulus of the zeta function on the critical line assuming RH, by providing explicit lower-order terms.

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Emanuel Carneiro, Micah B. Milinovich. 2024-03-26. On Littlewood's estimate for the modulus of the zeta function on the critical line. https://arxiv.org/abs/2403.17803

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