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Sharp bounds for joint moments of the Riemann zeta function

Michael J. Curran, André Heycock

Abstract

In previous work, the first author obtained conjecturally sharp upper bounds for the joint moments of the $(2k-2h)^{\text{th}}$ power of the Riemann zeta function with the $2h^{\text{th}}$ power of its derivative on the critical line in the range $1\leq k \leq 2$, $0 \leq h \leq 1$. Unconditionally, we extend these upper bounds to all $0 \leq h\leq k \leq 2$, and obtain lower bounds for all $0\leq h \leq k+1/2$. Assuming the Riemann hypothesis, we give sharp bounds for all $0\leq h \leq k$. We also prove upper bounds of the conjectured order for more general joint moments of zeta with its higher derivatives.

Sharp bounds for joint moments of the Riemann zeta function

Abstract

In previous work, the first author obtained conjecturally sharp upper bounds for the joint moments of the power of the Riemann zeta function with the power of its derivative on the critical line in the range , . Unconditionally, we extend these upper bounds to all , and obtain lower bounds for all . Assuming the Riemann hypothesis, we give sharp bounds for all . We also prove upper bounds of the conjectured order for more general joint moments of zeta with its higher derivatives.
Paper Structure (9 sections, 5 theorems, 40 equations)

This paper contains 9 sections, 5 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Lower Bounds: Proof of Theorem \ref{['thm:lowerJM']}
  3. Upper Bounds: Proof of Theorem \ref{['thm:mainRH']} for $0\leq k \leq 2$
  4. Initial reduction
  5. Notation
  6. Reduction to twisted moments
  7. Twisted fourth moment bound for $\zeta^{(n)}(\tfrac{1}{2}+it)$
  8. Applying the moment bounds
  9. Upper Bounds: Proof of Theorem \ref{['thm:mainRH']} for $k\geq \tfrac{1}{2}$

Key Result

Theorem 1.1

Let $n\in\mathbb Z_{>0}$, $0\leq k \leq 2$, and $h_j \geq 0$ for $1\leq j \leq n$, with $h:=h_1 + \cdots + h_n \leq k$. Then unconditionally for $k\leq 2$, and under the assumption of the Riemann hypothesis for $k>2$.

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof