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Lower bounds for the large deviations and moments of the Riemann zeta function on the critical line

Louis-Pierre Arguin, Nathan Creighton

Abstract

Building on work in \cite{AB24} on the Riemann zeta function at height $T$ off the critical line, we prove an unconditional lower bound on the critical line for real large deviations of the order $V\simα\log\log T$ for any $α>0.$ This gives another proof of the sharpest known unconditional lower bounds on the fractional moments of the Riemann zeta function, due to \cite{HSlower}. The lower bound on large deviations is of the same order of magnitude as the upper bound proved in \cite{AB23}, for the range $0<α<2.$

Lower bounds for the large deviations and moments of the Riemann zeta function on the critical line

Abstract

Building on work in \cite{AB24} on the Riemann zeta function at height off the critical line, we prove an unconditional lower bound on the critical line for real large deviations of the order for any This gives another proof of the sharpest known unconditional lower bounds on the fractional moments of the Riemann zeta function, due to \cite{HSlower}. The lower bound on large deviations is of the same order of magnitude as the upper bound proved in \cite{AB23}, for the range
Paper Structure (15 sections, 16 theorems, 150 equations)

This paper contains 15 sections, 16 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Results in the $q$-aspect
  4. Organisation of the proof
  5. Notation
  6. Acknowledgements
  7. Method of Proof
  8. The partial sums and their restrictions
  9. Proof of Theorem \ref{['LBLD']}
  10. Proof of Proposition \ref{['prop:goodprobs']}
  11. Approximation of the events $G_{\mathscr{J}}$
  12. Twisted moments for real partial sums
  13. Estimating restricted expectations
  14. Second and fourth twisted mollifier formulae
  15. Probabilities in the Steinhaus model

Key Result

Theorem 1.1

Let $\alpha>0$ and $V\sim \alpha \log\log T.$ Then for $T$ sufficiently large, we have with the constant $K_\alpha$ satisfying as $\alpha\to\infty,$ for some absolute constant $C$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2: Lemma 2 in AB24
  • ...and 16 more