Amplified moments of the Riemann zeta function

Technical Summary: Amplified Moments of the Riemann Zeta Function

Problem Statement
The evaluation of moments of the Riemann zeta function, defined as $M_k(T) = \int_0^T |\zeta(1/2 + it)|^{2k} dt$, is a central problem in analytic number theory. While asymptotic formulae are known for $k=1$ and $k=2$ (Hardy–Littlewood and Ingham), and sharp bounds exist for higher $k$ (conditional on the Riemann Hypothesis for upper bounds), unconditional asymptotic formulae for $k \ge 3$ remain out of reach. The primary objective of this work is to establish improved, unconditional effective lower bounds for the classical moments $M_k(T)$ and various joint moments involving derivatives of $\zeta(s)$. These bounds are compared against the conjectures of Keating–Snaith, Keating–Wei, and others, which are motivated by random matrix theory.

Methodology
The authors employ a method of "amplified moments," utilizing twisted mean values to probe higher moments. The core technique involves Hölder's inequality applied to integrals of the form:
$$\int_0^T |\zeta(1/2 + it)|^{2k} dt \ge \frac{\left( \int_0^T |\zeta(1/2 + it)|^4 |A(1/2 + it) + \chi(1/2 + it)A(1/2 - it)|^{2k-4} dt \right)^{k-1}}{\left( \int_0^T |\zeta(1/2 + it)|^2 |A(1/2 + it) + \chi(1/2 + it)A(1/2 - it)|^{2k-2} dt \right)^{k-2}}$$
where $A(s)$ is a Dirichlet polynomial of length $y = T^\theta$ (or $T^\vartheta$ for fourth moments) with smooth polynomial coefficients $P[n]$.

Key methodological components include:

1. Two-Piece Amplifier: Unlike previous approaches that used a single Dirichlet polynomial, the authors utilize the "two-piece" amplifier $A(s) + \chi(s)A(1-s)$, which mimics the approximate functional equation. This allows for better modeling of $\zeta(s)$ and facilitates the handling of joint moments.
2. Binomial Decomposition: The power of the amplifier is expanded using the Binomial Theorem, separating the integral into a main term (diagonal contributions) and oscillatory terms involving powers of $\chi(1-s)$.
3. Asymptotic Evaluation: The authors establish asymptotic formulae for the twisted second and fourth moments with high powers of the amplifier. They prove that terms with high powers of $\chi(1-s)$ (specifically $\nu \ge 2$ for second moments and $\nu \ge 3$ for fourth moments) are negligible ($O(T^{1-\varepsilon})$) due to rapid oscillation, provided the amplifier length parameters satisfy $\theta_k < 1/(2k)$ and $\vartheta_k < 1/(4k)$.
4. Polytope Integrals: The main terms are reduced to high-dimensional integrals over polytopes (simplexes). These integrals are evaluated explicitly using Mellin transforms and properties of the Riemann zeta function, resulting in expressions involving the Barnes $G$-function and arithmetic factors.
5. Optimization: The coefficients of the polynomial $P(x)$ are optimized numerically (using Wolfram Mathematica) to maximize the resulting lower bounds.

Key Contributions and Results

• Unconditional Lower Bounds for Classical Moments:
  The paper establishes a new unconditional lower bound for the sixth moment:
  $$M_3(T) \ge (34.1 + o(1))c_3 T (\log T)^9$$
  This improves upon previous bounds by Conrey–Ghosh, Soundararajan, and Page, approaching the Keating–Snaith conjectured value of $42c_3 T (\log T)^9$.

• Derivative Moments:
  New unconditional lower bounds are obtained for the sixth moments of the first and second derivatives of $\zeta(s)$:
  $$\int_0^T |\zeta'(1/2 + it)|^6 dt \ge (0.549 + o(1))c_3 T (\log T)^{15}$$
  $$\int_0^T |\zeta''(1/2 + it)|^6 dt \ge (0.0231 + o(1))c_3 T (\log T)^{21}$$
  These results improve upon previous bounds by Page and others.

• Joint Moments:
  The authors provide unconditional lower bounds for several joint moments involving $\zeta(s)$ and its derivatives (e.g., $\int |\zeta|^2 |\zeta'|^4$, $\int |\zeta|^4 |\zeta'|^2$, etc.). These bounds are in strong agreement with the conjectures of Keating–Wei.

• Refinement of Soundararajan's Bounds (Theorem 10):
  The paper removes the assumption of the Lindelöf Hypothesis from Soundararajan's bounds for $M_5(T)$ and $M_6(T)$, making them unconditional. Furthermore, by introducing a second piece to the amplifier, the authors improve the lower bounds for $M_k(T)$ for $7 \le k \le 11$. For example:
  $$M_5(T) \ge (6484 + o(1))c_5 T (\log T)^{25}$$
  $$M_6(T) \ge (56260 + o(1))c_6 T (\log T)^{36}$$
  These results are derived by modifying Soundararajan's arguments to handle the oscillatory terms unconditionally.

• General Joint Moments (Theorem 11):
  A general unconditional lower bound is established for all joint integer moments of $\zeta(s)$ and its derivatives. The bound is given in terms of a constant $C(a_1, \dots, a_K)$ derived from an Euler product and a derivative of a rational function, consistent with predictions from random matrix theory.

Significance and Claims
The paper claims to significantly advance the state of knowledge regarding lower bounds for zeta moments. Its primary significance lies in:

1. Removing Conditional Assumptions: It successfully removes the Lindelöf Hypothesis requirement for specific high-order moments ($k=5, 6$) and provides unconditional bounds for a wide range of joint moments.
2. Improving Numerical Constants: The derived lower bounds are numerically closer to the conjectured values (Keating–Snaith, Keating–Wei) than any previous unconditional results.
3. Methodological Flexibility: The "two-piece" amplified moment approach is shown to be particularly effective for joint moments, overcoming computational difficulties associated with polytope integrals for higher moments by combining them with Soundararajan's amplifier techniques.

The authors note that while their method yields strong bounds, the length of the permissible Dirichlet polynomial decreases as the moment order $k$ increases, which limits the recovery of certain uniform phenomena observed in other frameworks (such as those by Bui, Hall, and Subira Jorge). However, for the specific goal of obtaining effective unconditional lower bounds for integer moments, the approach proves highly successful.