papers/lp-307-quantum-consensus-protocol/sections/03-convergence-bounds-for-truncated-eisenst.tex

\section{Convergence Bounds for Truncated Eisenstein Series}


\subsection{Classical Truncation Error}

The tail of the Eisenstein series satisfies a standard bound:

\begin{theorem}[Truncation Bound]\label{thm:truncation}
For $k \geq 4$ and $z \in \mathcal{H}$ with
$\operatorname{Im}(z) \geq \delta > 0$:
\[
\left| G_k(z) - \tilde{G}_k^{(N)}(z) \right|
\leq C_k \cdot \delta^{-k} \cdot N^{2-k}
\]
where $C_k = 4\zeta(k-1)$ and $\zeta$ is the Riemann zeta function.
\end{theorem}

\begin{proof}[Proof sketch]
For fixed $c \neq 0$, the standard lattice-point counting bound
gives $\sum_{d \in \mathbb{Z}} |cz+d|^{-k}
   = O\!\left( |c \operatorname{Im}(z)|^{1-k} \right)$
(see Zagier~\cite{zagier2008}, Section 1.5). Summing over $|c| > N$
and applying the integral comparison
$\sum_{|c|>N} |c|^{1-k} = O(N^{2-k})$
yields the claimed $O(\delta^{-k} N^{2-k})$ bound, with constant
factor $C_k = 4\zeta(k-1)$ from the inner $|c|$ summation. Full
derivation in~\cite{zagier2008}, Theorem 4.2.
\end{proof}

\subsection{Quantum Estimation Error}

Combining the truncation bound with quantum phase estimation error
$\epsilon_{\mathrm{QPE}}$, the total error for the quantum evaluation
of $G_k(z)$ is:

\begin{equation}\label{eq:total-error}
\left| G_k(z) - \hat{G}_k^{(N)}(z) \right|
\leq C_k \delta^{-k} N^{2-k} + \epsilon_{\mathrm{QPE}}
\end{equation}

To achieve total error $\epsilon$, set $N = \Theta((\epsilon
\delta^k)^{1/(k-2)})$ and $\epsilon_{\mathrm{QPE}} = \epsilon/2$.
The quantum circuit depth is
$O(N^2 \cdot k \cdot p \log p / \epsilon)$.

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