[Paper Review] The Laplace and Mellin transforms of powers of the Riemann zeta-function

Aleksandar Ivić|ArXiv.org|May 29, 2006

Analytic Number Theory Research23 references3 citations

TL;DR

This paper investigates the Laplace and Mellin transforms of powers of the Riemann zeta-function on the critical line, establishing new bounds for the $L_k(s)$ and $\mathcal{Z}_k(s)$ transforms. It derives key estimates linking these transforms to the $2k$-th moments of $|\zeta(1/2 + it)|$, proving $\int_0^T |\zeta(1/2 + it)|^{12} dt \ll_{\varepsilon} T^{2+\varepsilon}$, which is optimal up to the $\varepsilon$-loss.

ABSTRACT

This paper gives a survey of known results concerning the Laplace transform $$ L_k(s) := \int_0^\infty |ζ(1/2+ ix)|^{2k}{ m e}^{-sx}{ m d} x \qquad(k \in N, \R s > 0), $$ and the (modified) Mellin transform $$ {\cal Z}_k(s) := \int_1^\infty|ζ(1/2+ ix)|^{2k}x^{-s}{ m d} x\qquad(k\in N), $$ where the integral is absolutely convergent for $\R s \ge c(k) > 1$. Also some new results on these integral transforms of $|ζ(1/2+ ix)|^{2k}$ are given, which have important connections with power moments of the Riemann zeta-function $ζ(s)$.

Motivation & Objective

• To analyze the Laplace transform $L_k(s) = \int_0^\infty |\zeta(1/2 + ix)|^{2k} e^{-sx} dx$ and the modified Mellin transform $\mathcal{Z}_k(s) = \int_1^\infty |\zeta(1/2 + ix)|^{2k} x^{-s} dx$ for $k \in \mathbb{N}$.
• To connect these integral transforms to the power moments $I_k(T) = \int_0^T |\zeta(1/2 + it)|^{2k} dt$ via asymptotic estimates.
• To establish new bounds for the $2k$-th moments of $|\zeta(1/2 + it)|$, particularly for $k=6$, using spectral theory and explicit formulas.
• To prove that $\int_0^T |\zeta(1/2 + it)|^{12} dt \ll_{\varepsilon} T^{2+\varepsilon}$, which is optimal up to the $\varepsilon$-loss.

Proposed method

• The Laplace transform $L_k(s)$ is analyzed via integration by parts and bounds on $I_k(T)$, linking $L_k(1/T)$ to $I_k(T)$.
• The modified Mellin transform $\mathcal{Z}_k(s)$ is used in conjunction with inverse Mellin transforms and integral representations involving $F(s)$, the Mellin transform of a smooth cutoff function.
• Explicit spectral formulas from Motohashi are applied, particularly involving sums over Maass forms: $S_m(K;K',t) = \sum_{K < \kappa_j \leq K'} \alpha_j H_j^m(1/2) \cos(\kappa_j \log(4et/\kappa_j))$.
• The method relies on bounding $L^2$-norms of spectral sums $S_m(K;K',t)$, using $\int_T^{2T} |S_m(K;K',t)|^2 dt \ll_{\varepsilon} T^{1+\varepsilon} K^3$ for $m=1,2,3$.
• Partial summation and truncation techniques are used to handle the decay of coefficients $\kappa_j^{-3/2}$ and $\exp(-(\Delta \kappa_j / T)^2)$.
• The final bound is obtained by summing over dyadic intervals and using the relation $E_2(t) \sim \sum \alpha_j H_j^3(1/2) \kappa_j^{-3/2} \cos(\cdots)$, with $\int_0^T E_2^2(t) dt \ll_{\varepsilon} T^{2+\varepsilon}$.

Experimental results

Research questions

• RQ1 How do the Laplace and Mellin transforms of $|\zeta(1/2 + ix)|^{2k}$ relate to the power moments $I_k(T)$ of the Riemann zeta-function?
• RQ2 Can the $\mathcal{Z}_k(s)$ transform be used to derive nontrivial bounds for $\int_0^T |\zeta(1/2 + it)|^{2k} dt$?
• RQ3 What is the sharpest known bound for the 12th moment $\int_0^T |\zeta(1/2 + it)|^{12} dt$?
• RQ4 How do spectral theory and explicit formulas for Maass forms contribute to estimating high moments of $|\zeta(1/2 + it)|$?
• RQ5 To what extent can the $L_k(s)$ transform be used to derive bounds on $I_k(T)$ via Laplace transform inversion?

Key findings

• The Laplace transform $L_k(s)$ satisfies $L_k(\sigma) \ll_{\varepsilon} (1/\sigma)^{c_k + \varepsilon}$ as $\sigma \to 0^+$, which implies $I_k(T) \ll_{\varepsilon} T^{c_k + \varepsilon}$.
• The modified Mellin transform $\mathcal{Z}_k(s)$ is absolutely convergent for $\Re(s) \geq c(k) > 1$, and is used to derive moment estimates via inverse Mellin transforms.
• The spectral sum $S_m(K;K',t)$ satisfies $\int_T^{2T} |S_m(K;K',t)|^2 dt \ll_{\varepsilon} T^{1+\varepsilon} K^3$ for $m=1,2,3$, which is a key technical tool.
• The bound $\int_0^T |\zeta(1/2 + it)|^{12} dt \ll_{\varepsilon} T^{2+\varepsilon}$ is established, which is optimal up to the $\varepsilon$-loss.
• The bound $\int_0^T E_2^2(t) dt \ll_{\varepsilon} T^{2+\varepsilon}$ is proven, where $E_2(t)$ is the error term in the fourth moment formula, and this implies the 12th moment bound.
• The result $\int_0^T |\zeta(1/2 + it)|^{12} dt \ll_{\varepsilon} T^{2+\varepsilon}$ is a consequence of the spectral sum bound and dyadic summation over $t \sim 2^{-j}T$.