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

Emanuel Carneiro, Micah B. Milinovich

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.

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

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.
Paper Structure (15 sections, 5 theorems, 61 equations)

This paper contains 15 sections, 5 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminary lemmas
  3. Extremal bandlimited approximations
  4. Explicit formula
  5. Estimates for zeta on the critical strip
  6. First term
  7. Second term
  8. Third term
  9. Fourth term
  10. Conclusion
  11. Proof of Theorem \ref{['new_Sound']}
  12. Proof of Theorem \ref{['lower-order']}
  13. Setup
  14. Choosing $x$: initial approximations
  15. The optimal choice of $x$

Key Result

Theorem 1

Assume RH. For $t\ge 10$ and $x\ge 2$, we have

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Lemma 3: Extremal functions for the Poisson kernel
  • Lemma 4: Guinand-Weil explicit formula
  • proof
  • Lemma 5
  • proof