Let \(\Delta_g\) be the Laplacian of a closed Riemannian manifold \((M^n,g)\). Its spectrum can be arranged as a non-decreasing sequence,

\[ \lambda_0(\Delta_g) \leq \lambda_1(\Delta_g) \leq \lambda_2(\Delta_g) \leq \cdots, \]

where each eigenvalue is repeated according to multiplicity. Its zeta function is defined by

\[ \zeta(\Delta_g;s) := \sum \lambda_j(\Delta_g)^{-s}, \qquad \operatorname{Re} s > \frac{n}{2}. \]

It has a meromorphic extension to \(\mathbb{C}\) with at worst simple poles on

\[ \Sigma = \left\{ \frac{n}{2}-j; \ j=0,1,2,\ldots \right\}. \]

Question. Given any \(\delta \in (0,1)\), can we find \(\kappa > 0\) such that

\[ \left| \zeta\bigl(\Delta_g; n/2-\delta + iy\bigr) \right| = \operatorname{O}\left(|y|^\kappa\right) \qquad \text{as } |y|\rightarrow \infty? \]

Let $\Delta_g$ be the Laplacian of a closed Riemannian manifold $(M^n,g)$. Its spectrum can be arranged as a non-decreasing sequence,
\[
 \lambda_0(\Delta_g)\leq \lambda_1(\Delta_g) \leq \lambda_2(\Delta_g) \leq \cdots,
\]
where each eigenvalue is repeated according to multiplicity. Its zeta function is defined by
\[
 \zeta(\Delta_g;s):= \sum \lambda_j(\Delta_g)^{-s}, \qquad \operatorname{Re} s>\frac{n}{2}.
\]
It has a meromorphic extension to $\mathbb C$ with at worst simple poles on $\Sigma=\left\{ \frac{n}{2}-j; \ j=0,1,2,\ldots\right\}$.

\textbf{Question.}
Given any $\delta\in(0,1)$, can we find $\kappa>0$ such that
\[
 \left|\zeta\big(\Delta_g; n/2-\delta +iy\big)\right| = \operatorname{O}\left(|y|^\kappa\right)\qquad \text{as}\ |y|\rightarrow \infty?
\]