Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Correlations of the squares of the Riemann zeta on the critical line

Valeriya Kovaleva

Abstract

We compute the average of a product of two shifted squares of the Riemann zeta on the critical line with shifts up to size $T^{3/2-\varepsilon}$. We give an explicit expression for such an average and derive an approximate spectral expansion for the error term similar to Motohashi's. As a consequence, we also compute the $(2,2)$-moment of moment of the Riemann zeta, for which we partially verify (and partially refute) a conjecture of Bailey and Keating.

Correlations of the squares of the Riemann zeta on the critical line

Abstract

We compute the average of a product of two shifted squares of the Riemann zeta on the critical line with shifts up to size . We give an explicit expression for such an average and derive an approximate spectral expansion for the error term similar to Motohashi's. As a consequence, we also compute the -moment of moment of the Riemann zeta, for which we partially verify (and partially refute) a conjecture of Bailey and Keating.
Paper Structure (24 sections, 36 theorems, 455 equations)

This paper contains 24 sections, 36 theorems, 455 equations.

Table of Contents

  1. Introduction
  2. Shifted averages
  3. Moments of moments
  4. Outline of the paper
  5. General moment conjectures and Motohashi's integral
  6. General conjecture
  7. Motohashi's integral
  8. Basic spectral theory
  9. Correlations of the divisor function
  10. Technical lemmas
  11. Groundwork
  12. Approximate functional equation
  13. Diagonal term
  14. Off-diagonal term for small shifts
  15. Transforming the integral weight
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $T \ge 10$ be large, let $\alpha,\beta \ge 0$ be such that $\delta := \beta-\alpha \ge 0$. Let $\gamma_0$ be the Euler-Mascheroni constant, and set Then $\mathcal{M}_4(T; \alpha,\beta) = \int_1^T (D(t;\alpha,\beta) + OD(t;\alpha,\beta))dt + E(T;\alpha,\beta)$, where and $E(T;\alpha,\beta) \ll O((T+\alpha)^{2/3+\varepsilon} + (T+\alpha)^{1/2+\varepsilon}\delta^{1/3} + \delta^{1/2})$.

Theorems & Definitions (62)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Conjecture 1.4: Bailey and Keating
  • Corollary 1.5
  • Corollary 1.6
  • Remark 1.7
  • Conjecture 2.1: CFKRS conrey2005integral
  • Theorem 2.2: Motohashi motohashi1994binary
  • Theorem 2.3: Motohashi motohashi1994binary
  • ...and 52 more