Consider a circle in $\mathbb{R}^2$, of radius $\rho > 0$, centred at the origin. Suppose that the number of integer points inside the circle is given by the function $N_{\rho}$. We then define the remainder (or error) to be
$$\displaystyle R_{\rho} = N_{\rho} - \pi \rho^2,$$
and it is conjectured that $R_{\rho} = O(\rho^{\frac{1}{2} + \epsilon})$, for any $\epsilon > 0$. In particular, it has been shown that we cannot take $\epsilon = 0.$ Now define the same function for a ball centred at $\mathbf{k} \in \mathbb{R}^d$ of radius $\rho$, so that:
$$\displaystyle R_{\rho}(\mathbf{k}) = N_{\rho}(\mathbf{k}) - \omega_d \rho^d,$$
where $\omega_d$ is the volume of the unit ball in $\mathbb{R}^d$.
Now, it is known that $R_{\rho}$ has Fourier co-efficients given by
$$\displaystyle \hat{R_{\rho}}(\mathbf{b}) = \left(\frac{2 \pi \rho}{|\mathbf{b}|} \right)^{d/2}J_{d/2}(\rho |\mathbf{b}|), \ \mathbf{b} \neq 0,$$
and $\hat{R_\rho}(0) = 0.$ Then, using the Fourier series of $R_{\rho}$, together with the estimates $|J_{\nu}(z)| \leqslant C_{\nu}|z|^{-1/2}$ and $|n| \leqslant |n|^2$ for any $n \in \mathbb{Z}^d \backslash \{0\}$, we have:
$$\displaystyle |R_{\rho}(\mathbf{k})| = \left| \sum_{\mathbf{b} \in \mathbb{Z}^d} \hat{R_{\rho}}(\mathbf{b})e^{i\mathbf{b}\mathbf{k}}\right| \leqslant \sum_{\mathbf{b} \in \mathbb{Z}^d} |\hat{R_{\rho}}(\mathbf{b})| = \sum_{\substack{\ \ \mathbf{b} \in \mathbb{Z}^d} \\ {\ \ \ \mathbf{b} \neq 0}} \left( \frac{2\pi \rho}{|\mathbf{b}|} \right)^{d/2} \left| J_{d/2}(\rho |\mathbf{b}|)\right| \\ \leqslant C_d (2 \pi)^{d/2} \rho^{\frac{d-1}{2}} \sum_{\substack{\ \ \mathbf{b} \in \mathbb{Z}^d} \\ {\ \ \ \mathbf{b} \neq 0}}|\mathbf{b}|^{-\frac{d+1}{2}} \leqslant C_d (2 \pi)^{d/2} \rho^{\frac{d-1}{2}} \sum_{\substack{\ \ \mathbf{b} \in \mathbb{Z}^d} \\ {\ \ \ \mathbf{b} \neq 0}}|\mathbf{b}|^{-(d+1)} = C \rho^{\frac{d-1}{2}},$$
where the last sum converges due to this answer, and $C, C_d, C_\nu$ are positive constants. But then setting $\mathbf{k} = 0$ and $d = 2$, so that the result coincides with the original Gauss circle problem, we get $R_{\rho} = O(\rho^{\frac{1}{2}})$ -- and that is obviously wrong! Where is the mistake in the proof?
There must be some major error here: have I applied any of the bounds incorrectly? I would like to know this, as I am working on various problems which depend on this.
Note that the expression $\mathbf{ab}$ denotes the scalar product of $\mathbf{a}$ and $\mathbf{b}$ on $\mathbb{R}^d$.
One error is in your last inequality. It is not true that $$ \sum_{b\in\mathbb{Z}^d\setminus \{0\}}|b|^{-(d+1)/2}\leq \sum_{b\in\mathbb{Z}^d\setminus \{0\}}|b|^{-(d+1)}, $$ since $|b|^{-(d+1)/2}> |b|^{-(d+1)}$ whenever $|b|> 1$, which is the case for all but finitely many summands here. You said that this follows from the inequality $|n|\leq |n|^2$ for all $|n|\geq 1$, but this implies that $|n|^{-2}\leq |n|^{-1}$ for all $|n|\geq 1$, not that $|n|^{-1}\leq |n|^{-2}$.
Alternatively, you can just plug in $d=1$ to the above inequality to see the falseness, since then the left-hand side diverges but the right-hand side converges.