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Explicit $L^2$ bounds for the Riemann $ζ$ function

Daniele Dona, Harald A. Helfgott, Sebastian Zuniga Alterman

Abstract

Explicit bounds on the tails of the zeta function $ζ$ are needed for applications, notably for integrals involving $ζ$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $ζ$. Two approaches are followed, each giving the better result on a different range. The first one is inspired by the proof of the standard mean value theorem for Dirichlet polynomials. The second approach, superior for large $T$, is based on classical lines, starting with an approximation to $ζ$ via Euler-Maclaurin. Both bounds give main terms of the correct order for $0<σ\leq 1$ and are strong enough to be of practical use for the rigorous computation of improper integrals. We also present bounds for the $L^{2}$ norm of $ζ$ in $[1,T]$ for $0\leqσ\leq 1$.

Explicit $L^2$ bounds for the Riemann $ζ$ function

Abstract

Explicit bounds on the tails of the zeta function are needed for applications, notably for integrals involving on vertical lines or other paths going to infinity. Here we bound weighted norms of tails of . Two approaches are followed, each giving the better result on a different range. The first one is inspired by the proof of the standard mean value theorem for Dirichlet polynomials. The second approach, superior for large , is based on classical lines, starting with an approximation to via Euler-Maclaurin. Both bounds give main terms of the correct order for and are strong enough to be of practical use for the rigorous computation of improper integrals. We also present bounds for the norm of in for .

Paper Structure

This paper contains 23 sections, 28 theorems, 149 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Methods and results
  4. Classical foundations revisited
  5. $O$ and $O^*$ notation
  6. Bernoulli polynomials
  7. Euler-Maclaurin summation formula
  8. The Mellin transform
  9. The Gamma function
  10. Bounds on some sums
  11. Further results
  12. First approach: as in a mean value theorem
  13. Basic estimate
  14. An "optimal" choice of $g$
  15. A better choice of $g$ for $\Re(s)\in\left[\frac{1}{2},1\right]$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $0<\sigma\leq 1$. Then, the integral $\int_{T}^{\infty}\left|\frac{\zeta(\sigma+it)}{\sigma+it}\right|^{2}dt$ is bounded as follows

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: MV07
  • Theorem 2.2: Euler-Maclaurin
  • Corollary 2.3
  • Lemma 2.4: Improved Euler-Maclaurin summation formula of second order
  • proof
  • Lemma 2.5
  • Theorem 2.6: Stirling's formula, explicit form
  • proof
  • ...and 39 more