TITLE: Reference for another equivalent of Riemann hypothesis
QUESTION [1 upvotes]: I don't know why but i assumed that $\ln \theta(x) + E < \sum \limits_{p \leq x} \frac{\ln p}{p} $ for all real numbers $x \geq 2$ and with $E = -\gamma - \sum \limits_{P} \frac{\ln p}{p(p-1)}  \approx -1.33258$ is equivalent to RH, i saw it somewhere
When i went back to find it in the articles in my laptop I just found the one directly related to Robin's inequality $\ln \ln \theta(x) +B < \sum \limits_{p \leq x} \frac{1}{p} $ for all $x\geq 3$.
Is there a ref talking about the first inequality being equivalent to RH, or a direct proof of it from Robin,Nicolas inequality ?
Thanks.

REPLY [2 votes]: Define von Mangold function as $\Lambda(n)=\log p[n=p^k]$ then
$$
\sum_{n\le x}{\Lambda(n)\over n}<\sum_{p\le x}\log p\sum_{k\ge1}p^{-k}=\sum_{p\le x}{\log p\over p}+\sum_{p\le x}\log p\sum_{k\ge2}p^{-k}<\sum_{p\le x}{\log p\over p}+\sum_p{\log p\over p(p-1)}
$$
As a result, we can use the definition of the constant $E$ to get
$$
\sum_{n\le x}{\Lambda(n)\over n}+\gamma+E<\sum_{p\le x}{\log p\over p}
$$
If RH is equivalent to saying
$$
\log\vartheta(x)\le\sum_{n\le x}{\Lambda(n)\over n}+\gamma
$$
then your statement will be another equivalent form of RH.


text:
\newpage

\section{Orange Juice Experiment}
\label{sec:oj}

Dominick's orange juice dataset (provided by the University of Chicago Booth School of Business) contains 28,947 entries of store-level, weekly prices and sales of different brands of orange juice. The dataset also contains 15 continuous and categorical variables that encode store-level customer information such as the mean age, income, education level, etc, as well as brand information. The goal is to learn the elasticity of orange juice as a function of income (or education, etc) in the presence of high-dimensional controls.

In the experiment depicted in Figure~\ref{fig:oj}, we trained the ORF using $500$ trees, a minimum leaf size of $50$, subsample ratio of $0.02$, with Lasso models for both residualization and kernel estimation. We evaluated the resulting algorithm on $50$ $\log(Income)$ points between $10.4$ and $10.9$. We then followed-up with 100 experiments on bootstrap samples of the original dataset to build bootstrap confidence intervals. The emerging trend in the elasticity as a function of income follows our intuition: higher income levels correspond to a more inelastic demand.

\section{All Experimental Results}
\label{sec:allexp}

We present all experimental results for the parameter choices
described in Section \ref{sec:mcexp}. We vary the support size $k\in\{1,5,10,15,20,25,30\}$,
the dimension $d\in\{1,2\}$ of the feature vector $x$ and the
treatment response function
$\theta \in\{\text{piecewise linear, piecewise constant and piecewise
  polynomial}\}$. We measure the bias, variance and root mean square
error (RMSE) as evaluation metrics for the different estimators we
considered in Section \ref{sec:mcexp}. In addition, we add another version of the GRF (GRF-xW) where we run the GRF R package directly on the observations, using features and controls $(x, W)$ jointly as the covariates. For the parameter space we consider, the ORF-CV and the ORF
algorithms outperform the other estimators on all regimes.

\subsection{Experimental results for one-dimensional, piecewise linear $\theta_0$}
Consider a piecewise linear function: $
\theta_0(x) = (x+2)\mathbb{I}_{x\leq 0.3} + (6x+0.5)\mathbb{I}_{x>0.3 \text{ and } x \leq 0.6} + (-3x+5.9)\mathbb{I}_{x> 0.6}$.

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_linear_postprocess_results/pdf_low_res/Metrics_by_support.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{Bias, variance and RMSE as a function of support size for $n=5000$, $p=500$, $d=1$ and $\theta_0$. The solid lines represent the mean of the metrics across test points, averaged over the 100 experiments, and the filled regions depict the standard deviation, scaled down by a factor of 3 for clarity.}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_linear_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_1_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
		Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=1}$, and a piecewise linear treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_linear_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_5_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=5}$, and a piecewise linear treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_linear_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_10_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
		Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=10}$, and a piecewise linear treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_linear_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_15_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=15}$, and a piecewise linear treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_linear_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_20_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=20}$, and a piecewise linear treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_linear_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_25_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=25}$, and a piecewise linear treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
		\begin{minipage}[c]{0.75\textwidth}
			\includegraphics[width=\textwidth]{figures/piecewise_linear_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_30_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
		\end{minipage}\hfill
		\begin{minipage}[c]{0.23\textwidth}
			\caption{
				Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=30}$, and a piecewise linear treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
			}
		\end{minipage}
	\end{figure}

\pagebreak
\subsection{Experimental results for one-dimensional, piecewise constant $\theta_0$}

We introduce the results for a piecewise constant function given by:
\begin{align*}
\theta_0(x) &= \mathbb{I}_{x\leq 0.2} + 5\mathbb{I}_{x>0.2 \text{ and } x \leq 0.6} + 3\mathbb{I}_{x> 0.6}
\end{align*}

\begin{figure}[H]
	\centering
	\includegraphics[scale=.50]{figures/step_postprocess_results/pdf_low_res/Metrics_by_support.pdf}
	\caption{Bias, variance and RMSE as a function of support size for $n=5000$, $p=500$, $d=1$ and a piecewise constant treatment function. The solid lines represent the mean of the metrics across test points, averaged over the Monte Carlo experiments, and the filled regions depict the standard deviation, scaled down by a factor of 3 for clarity.}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/step_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_1_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=1}$, and a piecewise constant treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/step_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_5_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=5}$, and a piecewise constant treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\pagebreak

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/step_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_10_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=10}$, and a piecewise constant treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/step_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_15_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=15}$, and a piecewise constant treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/step_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_20_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=20}$, and a piecewise constant treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\pagebreak

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/step_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_25_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=25}$, and a piecewise constant treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
		\begin{minipage}[c]{0.75\textwidth}
			\includegraphics[width=\textwidth]{figures/step_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_30_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
		\end{minipage}\hfill
		\begin{minipage}[c]{0.23\textwidth}
			\caption{
				Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=30}$, and a piecewise constant treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
			}
		\end{minipage}
	\end{figure}

\pagebreak
\subsection{Experimental results for one-dimensional, piecewise polynomial $\theta_0$}

We present the results for a piecewise polynomial function given by:
\begin{align*}
\theta_0(x) &= 3x^2\mathbb{I}_{x\leq 0.2} + (3x^2+1)\mathbb{I}_{x>0.2 \text{ and } x \leq 0.6} + (6x+2)\mathbb{I}_{x> 0.6}
\end{align*}

\begin{figure}[H]
	\centering
	\includegraphics[scale=.50]{figures/poly_postprocess_results/pdf_low_res/Metrics_by_support.pdf}
	\caption{Bias, variance and RMSE as a function of support size for $n=5000$, $p=500$, $d=1$ and a piecewise polynomial treatment function. The solid lines represent the mean of the metrics across test points, averaged over the Monte Carlo experiments, and the filled regions depict the standard deviation, scaled down by a factor of 3 for clarity.}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/poly_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_1_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=1}$, and a piecewise polynomial treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/poly_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_5_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=5}$, and a piecewise polynomial treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\pagebreak

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/poly_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_10_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=10}$, and a piecewise polynomial treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/poly_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_15_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=15}$, and a piecewise polynomial treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/poly_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_20_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=20}$, and a piecewise polynomial treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\pagebreak

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/poly_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_25_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=25}$, and a piecewise polynomial treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
		\begin{minipage}[c]{0.75\textwidth}
			\includegraphics[width=\textwidth]{figures/poly_postprocess_results/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_30_n_trees_100_max_depth_20_min_leaf_size_5_s_pow_0_88_.pdf}
		\end{minipage}\hfill
		\begin{minipage}[c]{0.23\textwidth}
			\caption{
				Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=30}$, and a piecewise polynomial treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
			}
		\end{minipage}
	\end{figure}

\pagebreak
\subsection{Experimental results for larger control support}

We present experimental results for a piecewise linear treatment response $\theta_0$, with $n=5000$ samples and large support $k\in \{50,75,100,150,200\}$. Figures \ref{fig:large1}-\ref{fig:large2} illustrate that the behavior of the ORF-CV algorithm, with parameters set in accordance our theoretical results, is consistent up until fairly large support sizes. Our method performs well with respect to the chosen evaluation metrics and outperform other estimators for larger support sizes.

\begin{figure}[H]
	\centering
	\includegraphics[scale=.45]{figures/piecewise_large_support/pdf_low_res/Metrics_by_support.pdf}
	\caption{Bias, variance and RMSE as a function of support size. The solid lines represent the mean of the metrics across test points, averaged over the Monte Carlo experiments, and the filled regions depict the standard deviation, scaled down by a factor of 3 for clarity.}
	\label{fig:large1}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.70\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_large_support/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_50_n_trees_200_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.25\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=50}$, and a piecewise linear treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.70\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_large_support/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_75_n_trees_200_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.25\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=75}$, and a piecewise linear treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\pagebreak

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_large_support/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_100_n_trees_200_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=100}$, and a piecewise linear treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_large_support/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_150_n_trees_200_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=150}$, and a piecewise linear treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_large_support/pdf_low_res/Example_n_samples_5000_n_dim_500_n_x_1_support_200_n_trees_200_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=1$, $\mathbf{k=200}$, and a piecewise linear treatment response. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
		\label{fig:large2}
	\end{minipage}
\end{figure}

\newpage
\subsection{Experimental results for two-dimensional heterogeneity}

We introduce experimental results for a two-dimensional $x$ and corresponding $\theta_0$ given by:
\begin{align*}
\theta_0(x_1,x_2) &= \theta_{\text{piecewise linear}}(x_1)\mathbb{I}_{x_2=0} +
\theta_{\text{piecewise constant}}(x_1)\mathbb{I}_{x_2=1}
\end{align*}
where $x_1\sim U[0, 1]$ and $x_2\sim Bern(0.5)$. In Figures \ref{fig:2D_1}-\ref{fig:2D_2}, we examine the overall behavior of the ORF-CV and ORF estimators, as well as the behavior across the slices $x_2=0$ and $x_2=1$. We compare the performance of the ORF-CV and ORF estimators with alternative methods for $n=5000$ and $k\in\{1,5,10,15,20,25,30\}$. We conclude that the ORF-CV algorithm yields a better performance for all support sizes and evaluation metrics.

\begin{figure}[H]
	\centering
	\includegraphics[scale=.50]{figures/piecewise_2D/pdf_low_res/Metrics_by_support_x1=all.pdf}
	\caption{Overall bias, variance and RMSE as a function of support size for $n=5000$, $p=500$, $d=2$. The solid lines represent the mean of the metrics across test points, averaged over the Monte Carlo experiments, and the filled regions depict the standard deviation, scaled down by a factor of 3 for clarity.}
	\label{fig:2D_1}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.75\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Metrics_by_support_x1=0.pdf}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Metrics_by_support_x1=1.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.23\textwidth}
		\caption{
			Bias, variance and RMSE as a function of support for $n=5000$, $p=500$, $d=2$ and slices $\mathbf{x_2=0}$ and $\mathbf{x_2=1}$, respectively. The solid lines represent the mean of the metrics across test points, averaged over the Monte Carlo experiments, and the filled regions depict the standard deviation, scaled down by a factor of 3 for clarity.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.63\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=0_n_samples_5000_n_dim_500_n_x_2_support_1_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=1_n_samples_5000_n_dim_500_n_x_2_support_1_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.34\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=2$, $\mathbf{k=1}$, and slices $\mathbf{x_2=0}$ and $\mathbf{x_2=1}$, respectively. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.63\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=0_n_samples_5000_n_dim_500_n_x_2_support_5_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=1_n_samples_5000_n_dim_500_n_x_2_support_5_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.34\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=2$, $\mathbf{k=5}$, and slices $\mathbf{x_2=0}$ and $\mathbf{x_2=1}$, respectively. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.63\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=0_n_samples_5000_n_dim_500_n_x_2_support_10_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=1_n_samples_5000_n_dim_500_n_x_2_support_10_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.34\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=2$, $\mathbf{k=10}$, and slices $\mathbf{x_2=0}$ and $\mathbf{x_2=1}$, respectively. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.63\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=0_n_samples_5000_n_dim_500_n_x_2_support_15_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=1_n_samples_5000_n_dim_500_n_x_2_support_15_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.34\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=2$, $\mathbf{k=15}$, and slices $\mathbf{x_2=0}$ and $\mathbf{x_2=1}$, respectively. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.63\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=0_n_samples_5000_n_dim_500_n_x_2_support_20_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=1_n_samples_5000_n_dim_500_n_x_2_support_20_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.34\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=2$, $\mathbf{k=20}$, and slices $\mathbf{x_2=0}$ and $\mathbf{x_2=1}$, respectively. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.63\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=0_n_samples_5000_n_dim_500_n_x_2_support_25_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=1_n_samples_5000_n_dim_500_n_x_2_support_25_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.34\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=2$, $\mathbf{k=25}$, and slices $\mathbf{x_2=0}$ and $\mathbf{x_2=1}$, respectively. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
	\end{minipage}
\end{figure}

\begin{figure}[H]
	\begin{minipage}[c]{0.63\textwidth}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=0_n_samples_5000_n_dim_500_n_x_2_support_30_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
		\includegraphics[width=\textwidth]{figures/piecewise_2D/pdf_low_res/Example_x1=1_n_samples_5000_n_dim_500_n_x_2_support_30_n_trees_100_max_depth_40_min_leaf_size_25_s_pow_0_88_.pdf}
	\end{minipage}\hfill
	\begin{minipage}[c]{0.34\textwidth}
		\caption{
			Treatment effect estimations for 100 Monte Carlo experiments with parameters $n=5000$, $p=500$, $d=2$, $\mathbf{k=30}$, and slices $\mathbf{x_2=0}$ and $\mathbf{x_2=1}$, respectively. The shaded regions depict the mean and the $5\%$-$95\%$ interval of the 100 experiments.
		}
		\label{fig:2D_2}
	\end{minipage}
\end{figure}

config: arxiv


TITLE: Does a rotating disk develop a potential difference between the centre and rim?
QUESTION [10 upvotes]: This stems from thinking about the question If a perfect conductor were to move, what happens to the electrons?.
Suppose we have a rotating disk with no external magnetic field, so this is not a homopolar generator/Faraday disk experiment. The rotation creates a difference in potential energy between the centre and the rim, so does this mean that if we connect a wire (with suitable brushes) between the centre and the rim electrons will flow from the centre through the disk to the rim then back through the wire? That is, does the rotation create an electrical potential difference between the centre and the rim?
It seems obvious to me that the answer is yes, however I have never seen the calculation done. Attempts to Google it fail because the results are swamped by articles on Faraday disks and/or homopolar generators.

REPLY [1 votes]: This answer is valid under the assumption that the wire also rotates with the disk.
@cag's answer reveals two things :
(1) that the electric field is independent of the distribution of charge in the disk. We know that this distribution will vary for materials with different conductivities.
(2) that the field is independent of the shape of the conducter under rotation.
Since the wire is under rotation, a centripetal force exists in the wire as well. And thus, from @cag's formula we get to know that the potential difference in the wire is exactly same as that of the disk and in the same direction.
Now, since you have specified the conductivity of the wire, being zero, we know that the distribution of charge is markedly different from that of the disk.
For a wire having same conductivity as that of the disk, we know that no current would flow as soon as you connect the wire, since the net emf of the two cancel each other.
For a wire of different conductivity than that of the disk, the distribution difference between the two causes a kind of osmosis current which flows until it reaches a state where the distribution of charge along the radius of the disk is same for both.
Forgive me for my simplistic approach at solution, I'm not particularly adept at mechanisms on the microscopic scale.


text:
\begin{document}

\title
{Non-vanishing of  Hilbert Poincar\'e series}
\author[M. Kumari]{Moni Kumari}
\address{National Institute of Science Education and Research, HBNI, Bhubaneswar, Via-Jatni, Khurda, Odisha, 752050, India.}

\email{moni.kumari@niser.ac.in}

\date{\today}

\subjclass[2010]{Primary 11F41; Secondary 11F30.}

\keywords{Hilbert modular forms, Poincar\'e series, non-vanishing}
\begin{abstract}
We prove some non-vanishing results of Hilbert Poincar\'e series.
We derive these results, by showing that the Fourier coefficients of Hilbert Poincar\'e series satisfy some nice orthogonality relations for sufficiently large weight as well as for sufficiently large level.
To prove later results, we generalize a method of E. Kowalski et. al.
\end{abstract}
\maketitle

\section{Introduction}
The vanishing or non-vanishing of Poincar\'e series of integral weight for the group $SL_{2}(\mathbb{Z})$ is a mysterious problem. Here, there is a conjecture that none of the Poincar\'e series  vanish.
Several papers have appeared to investigate this conjecture. The first non-trivial result towards this question was given by R. A. Rankin \cite{rankin} in 1980, who showed that
there are constants $B>4\log{2}$ and $k_0$ such that the Poincar\'e series $\mathcal{P}_{n,k,1}(z)$ does not vanish identically for
\begin{equation}
n\leq k^2 \exp \bigg(\frac{-B\log k}{\log \log k}\bigg),
\end{equation}
provided $k\geq k_0,$ where $\mathcal{P}_{n,k,1}(z)$  is the $n$-th Poincar\'e series of integral weight $k$ and of level $1$.
His proof uses the fact that the Fourier coefficients of Poincar\'e series has an explicit formula as an infinite series involving Bessel functions and Kloosterman sums and sharp estimates for the magnitude of Kloosterman sums. Later, J. Lehner \cite{lehner} and C. J. Mozzochi \cite{mozz} generalized the Rankin's result for an arbitrary Fuchsian group and for the congruence subgroup $\Gamma_{0}(N)$ respectively. In 1986 E. Gaigalas \cite{gai}, using Weil's estimate for the Kloosterman sum proved the following. ``For any $m\in \mathbb{N}$ there exist infinitely many $k\in 2\mathbb{N}$ for which the $m$-th Poincar\'e series of weight $k$ (with respect to any finite index subgroup of $SL_2(\mathbb{Z})$) is not identically zero." It is notable that the non-vanishing of Poincar\'e series is related to the famous conjecture of Lehmer \cite{lehmer}, which says that $\tau(n)\neq 0$, for all $n\geq 1,$ where $\tau$ is the Ramanujan $\tau$-function.

As we know that the space of Hilbert modular forms is a generalization of the space of elliptic modular forms, it is natural to ask the vanishing or non-vanishing of Hilbert Poincar\'e series. To the best of our knowledge, in the literature there is no result concerning this problem. In this article, we give an answer for this question in terms of weight as well as level aspects. More precisely, we prove that for a fixed level $\mathcal{I}$ and for any finite set $\mathcal{A}\subset \mathcal{O}_F^*$, there exist a positive constant $k_{\mathcal{A}}$ such that for all $k\geq k_{\mathcal{A}}$ and for any $\nu \in\mathcal{A}$
\begin{equation}
\mathcal{P}_{\vec{k},\nu,\mathcal{I}}(z)\nequiv 0.
\end{equation}
See section \ref{notation} for the notations.
We also prove an analogous result with respect to level. Note that in a particular case we get a generalization of Gaigalas's result for Hilbert Poincar\'e series.

For the proof of our main results, we first prove (see \thmref{weight} and \thmref{level}) that the Fourier coefficients of Hilbert Poincar\'e series satisfy some nice orthogonality relations with respect to weight as well as level. These intermediate results generalize the work of E. Kowalski et. al. \cite{saha}. The main ingredients to prove these orthogonality relations are basic Fourier analysis for functions of several variables and dominated convergence theorem. These results may be of independent interest.

The paper is organized as follows. Section \ref{notation} contains some notations and a brief introduction about Hilbert modular forms. Here we also state the results of Kowalski et. al. \cite{saha} for Poincar\'e series. Furthermore, their method has been extended to Hilbert Poincar\'e series in section \ref{intermediate}. The main results of the paper which is about the non-vanishing of Hilbert Poincar\'e series are given in section \ref{mainresult}.
\section{Notations and preliminaries}\label{notation}
Let $F$ be a totally real number field of degree $n$ over $\mathbb{Q}$ and $\mathcal{O}_{F}$ be its ring of algebraic integers. Assume that $\sigma_1, \sigma_2,...,\sigma_n$ denote the real embeddings of $F$. We write $\alpha_i= \sigma _i(\alpha)$ for $\alpha \in F ~\mbox{and}~ 1\leq i\leq n.$ The trace and norm of $\alpha \in F$ are defined by ${\rm tr}(\alpha)=\sum_{i=1}^n \alpha_i$ and $N(\alpha)=\prod_{i=1}^n \alpha_i$ respectively. For $\alpha \in \mathbb{C}^n$, the trace and norm are defined by the sum and the product of its components respectively. More generally, if $ c= (c_1,c_2,\cdots, c_n), d=(d_1,d_2,\cdots, d_n), z=(z_1,z_2,\cdots,z_n)~ \mbox{and}~m=(m_1,m_2,\cdots,m_n)\in \mathbb{C}^n,$ then the norm and trace are define by
  \begin{equation*}
 N(cz+d):= \prod_{i=1}^{n}(c_iz_i+d_i) ~~~~~~~~{\rm ~and~~~}~~~~~~~~~{\rm tr}(mz):= \sum _{i=1}^{n}m_iz_i.
\end{equation*}
For $\alpha \in \mathcal{O}_{F},$ we write $\alpha \succeq 0$ to demonstrate that either $\alpha=0 $ or $\alpha$ is totally positive (means all the conjugates of $\alpha$ are positive) and $\alpha\gg 0$ for $\alpha$ to be totally positive.

Let $GL^+_{2}(\mathbb{R})$ be the set of all matrices in $GL_{2}(\mathbb{R})$ with positive determinant. We know that the group $GL^+_{2}(\mathbb{R})$ acts on the upper half plane
$\mathbb{H}=\{x+iy\in \mathbb C:y>0\}$ via,
$$\left(\begin{array}{cc}a&b\\c &d\end{array}\right)z=\frac{az+b}{cz+d}.$$
Now by using the above action, we define an action of $GL_{2}^+(\mathcal{O}_{F})$) on $\mathbb{H}^n$. Using all the embeddings of $F$ and fixing their order, we embed $GL_{2}(F)$ into $GL_{2}(\mathbb{R})^n.$ The image of $GL_{2}(\mathcal{O}_{F})$ in $GL_{2}(\mathbb{R})^n$ is discrete. We denote the set of elements in $GL_{2}(F)$ (respectively  $GL_{2}(\mathcal{O}_{F})$) with totally positive determinant by $GL^+_{2}(F)$ (respectively  $GL_{2}^+(\mathcal{O}_{F})$). For $g=(g_1,g_2,\cdots,g_n)\in GL_{2}^+(\mathbb{R})^n$ and $z=(z_1,z_2,\cdots,z_n)\in \mathbb{H}^n$, we define
$$ gz:=(g_1z_1,g_2z_2,\cdots,g_nz_n),$$
which deduces an action of $GL^+_{2}(F)$ and hence of $GL_{2}^+(\mathcal{O}_{F})$ on $\mathbb{H}^n.$

In this paper, we shall work with the Hilbert modular group
$$ \Gamma_{F}=SL_{2}(\mathcal{O}_{F}):= \bigg\{\left(\begin{array}{cc}a&b\\c &d\end{array}\right): a,b,c,d\in \mathcal{O} _{F},~ ad-bc=1 \bigg\}, $$
and its congruence subgroups of level $\mathcal{I}$, which is defined by
$$\Gamma_{0}(\mathcal{I}):= \bigg\{\left (\begin{array}{cc}\alpha&\beta\\ \gamma & \delta\end{array}\right) \in {\Gamma_{F}: \gamma \in \mathcal{I}}\bigg\},$$
where $\mathcal{I} \subseteq \mathcal{O} _{F}$ is a non-zero integral ideal.

Let $g=(g_1,g_2,\cdots,g_n)\in GL_{2}^+(\mathbb{R})^n$, $z=(z_1,z_2,\cdots,z_n)\in \mathbb{H}^n$, and $k=(k_1,k_2,\cdots,k_n)\in \mathbb{Z}^n$. We define
\begin{equation*}
\mu(g,z)^k:=\prod_{j=1}^{n}({\rm det}g_j)^{-k_j/2}(c_jz_j+d_j)^{k_j},
\end{equation*}
where $g_j=\left(\begin{array}{cc}*&*\\c_j &d_j\end{array}\right).$ Furthermore, for a function $f$ defined on $\mathbb{H}^n$, we set
$$(f|{_k}g)(z)=\mu(g,z)^{-k}f(gz).$$

\subsection{Hilbert modular forms}
A Hilbert modular form of weight $k \in \mathbb{N}^{n}_{0}$ for a congruence subgroup $ \Gamma$ of $\Gamma_{F}$ is a holomorphic function $f:\mathbb{H}^n \rightarrow \mathbb{C}$ such that
$$ f\mid_k \gamma= f,~\mbox{for~ all}~\gamma \in \Gamma.$$
For $n=1$, we also need holomorphicity condition at the cusps of $\Gamma$. Note that for $n>1$, a Hilbert modular form is automatically holomorphic at the cusps by
the Koecher principle \cite[Sec 1.4]{gar}. In addition, $f$ is called a Hilbert cusp form if it vanishes at all the cusps of $\Gamma.$ Let $M_{k}(\Gamma)$ denotes the space of Hilbert modular forms of  weight $k \in \mathbb{N}_{0}^{n}$ for the congruence subgroup $ \Gamma$ and $S_{k}(\Gamma)$ be the subspace of cusp forms. These are finite dimensional complex vector spaces and $S_{k}(\Gamma)$ is a Hilbert space with respect to the Petersson inner product
\begin{equation*}\label{inner}
\langle f,g\rangle_{} :=\int_{\Gamma\setminus \mathbb{H}^n} f(z)\overline{g(z)}y^k \frac{dxdy}{y^2},
\end{equation*}
where $z=x+iy, ~dx=dx_{1}\cdots dx_{n}~\mbox{and}~dy=dy_{1} \cdots dy_{n}.$
If $f\in M_{k}(\Gamma_{0}(\mathcal{I}))$, where $\mathcal{I} \subseteq \mathcal{O} _{F}$ is a non-zero integral ideal, then we call $f$ to be a Hilbert modular form of weight $k$ and of level $\mathcal{I}$. Note that if $F=\mathbb{Q}$, then $M_{k}(\Gamma)$ is the space of elliptic modular forms.

By the Koecher principle, $f\in M_{k}(\Gamma)$ has a Fourier expansion at the cusp $\infty$ of the form
\begin{equation*}
f(z)= \sum \limits_{\substack{m\in\Lambda_{\Gamma}^*\\m\succeq 0}} a_{m}e^{2\pi i {\rm tr}(m z)},
\end{equation*}
where
\begin{equation}\label{lambdaf}
\Lambda_{\Gamma}=\bigg\{\mu \in F:\left(\begin{array}{cc}1&\mu\\0 &1\end{array}\right)\in\Gamma \bigg\},~~
\Lambda_{\Gamma}^*=\big\{\mu\in F:{\rm tr}(\mu \Lambda_{\Gamma})\subseteq \mathbb{Z}\big\}.
\end{equation}
Here $\Lambda_{\Gamma}^*$ is called the dual space of $\Lambda_{\Gamma}$. Note that if $\Gamma=\Gamma_{0}(\mathcal{I})$ then $\Lambda_{\Gamma}=\mathcal{O}_{F}$ and $\Lambda_{\Gamma}^*=\mathcal{O}_{F}^*$.

Now we introduce some notations which will be used in later sections. For an integer $x \in\mathbb{N}_{0},$
 we denote $\vec{x}= (x,x, \cdots, x)\in \mathbb{N}_{0}^{n}.$ For $\nu= (\nu_{1},\nu_2,\cdots,\nu_{n}) \in \mathbb{N}_{0}^{n}~ \mbox{and}~ z=(z_1, z_2,\cdots, z_n)\in \mathbb{C}^n,$ we put
 $$ |\nu|= \sum_{i=1}^{n} \nu_{i},~\nu!= \prod_{i=1}^{n}\nu_{i}!~\mbox{and}~z^{\nu}=\prod_{i=1}^{n} z_{i}^{\nu_i}.$$
For an integral ideal $\mathcal{I}$ of $\mathcal{O}_F$, we denote the norm of  $\mathcal{I}$ by $\mathcal{N}(\mathcal{I}):=[\mathcal{O}_F:\mathcal{I}].$ For $z=(z_1, z_2,\cdots, z_n)\in \mathbb{H}^n,$ we always write ${\rm{Im}}(z)  =({\rm{Im}}(z_1), {\rm{Im}}(z_2),\cdots, {\rm{Im}}(z_n))$. Throughout the paper, we write Poincar\'e series for Poincar\'e series of the group $SL_2(\mathbb Z)$ and its congruence subgroups.

\subsection{Hilbert Poincar\'e Series:}
Let $\mathcal{I}\subseteq\mathcal{O}_{F}$ be a non-zero integral ideal and $\Gamma_{0}(\mathcal{I})$ be the associated congruence subgroup. For a totally positive element $\nu$ of $\mathcal{O}_{F}^*$ and weight $k=(k_1,k_2,\cdots,k_n)$ $(k_j>2, ~j=1,2,\cdots,n)$, we define the $\nu$-th Hilbert Poincar\'e series as follows:
\begin{equation}\label{pdefinition}
\mathcal{P}_{k,\nu,\mathcal{I}}(z)=\sum_{M\in\Gamma_{\infty}\setminus \Gamma_{0}(\mathcal{I})}\mu(M,z)^{-k}e^{2\pi i{\rm tr}(\nu(Mz))},
\end{equation}
where $\Gamma_{\infty}=\bigg\{\left(\begin{array}{cc}1& \mu \\0 &1\end{array}\right): \mu\in\mathcal{O} _{F}\bigg\}.$ It is well known that $\mathcal{P}_{k,\nu,\mathcal{I}} \in S_{k}(\Gamma_{0}(\mathcal{I}))$.

For more details on the theory of Hilbert modular forms, we refer \cite{fre} and \cite{gar}.
\subsection{Orthogonality of Fourier coefficients of Poincar\'e series} Here we recall some results obtained by Kowalski et. al. \cite{saha}, concerning the orthogonality properties of the Fourier coefficients of Poincar\'e series. In the next section we generalize them for Hilbert Poincar\'e series.

Let $k > 2$ and $m\geq 1$ be integers. Let $\mathcal{P}_{m,k,q}(z)$ be the $m$-th Poincar\'e series of weight $k$ for the group $\Gamma_{0}(q)$. Recently, Kowalski et. al. \cite{saha} proved that the Fourier coefficients $p_{m,k,q}(n)$ of the Poincar\'e series $\mathcal{P}_{m,k,q}(z)$ satisfy the following orthogonal relations with respect to weight $k$ as well as the level $q$.
\begin{prop}\label{k}
With notations as above, for fixed $m\geq 1$ and $n\geq 1$, we have
\begin{equation*}
\lim_{k\rightarrow\infty}p_{m,k,1}(n)=\delta(m,n),
\end{equation*}
where $\delta(\cdot, \cdot)$ is the Kronecker symbol.
\end{prop}
Furthermore, they also showed similar result with respect to another important parameter, the level $q$.

\begin{prop}\label{q}
With notations as above, for fixed $k\geq 4, m$ and $n$, we have
\begin{equation*}
\lim_{q\rightarrow \infty}p_{m,k,q}(n)=\delta(m,n).
\end{equation*}
\end{prop}
Note that the above orthogonality relations can be obtained by applying the trace formula to Poincar\'e series. But in \cite{saha}, the authors gave a completely different and a soft proof. Furthermore, in the same paper they have also obtained a similar orthogonality results for the coefficients of Siegel Poincar\'e series

\section{Intermediate results: Orthogonality relations}\label{intermediate}
In this section, we extend \propref{k} and \propref{q} for Hilbert Poincar\'e series, which will be used to get our main results. We remark that these results may be of independent interest.

Let $p_{k,\nu,\mathcal{I}}(\mu)$ be the $\mu$-th Fourier coefficient of the Hilbert Poincar\'e series $\mathcal{P}_{k,\nu,\mathcal{I}}$ defined by \eqref{pdefinition}, where $\mu\in\mathcal{O}_F^*.$ First, we prove that the Fourier coefficients $p_{k,\nu,\mathcal{I}}$ satisfy an orthogonality relation for sufficiently large $k.$ The main ingredient for the proof is dominated convergence theorem. In \thmref{weight}, we shall only work with parallel weight $\vec{k}=(k,k,...,k).$
\begin{thm}{\bf [Weight aspect]}\label{weight}
Let $\mathcal{P}_{\vec{k},\nu,\mathcal{I}}$ be the $\nu$-${\rm th}$ Hilbert Poincar\'e series of weight $\vec{k}$ and of level $\mathcal{I}$. Then for fixed $\nu\gg 0$ and  $\mu\gg 0$, we have
$$\lim_{k\rightarrow \infty}p_{\vec{k},\nu,\mathcal{I}}(\mu)= \delta(\nu, \mu).$$
\end{thm}

\begin{proof}
From the theory of Fourier analysis for function of several variables, we have
\begin{equation*}
 p_{\vec{k},\nu,\mathcal{I}}(\mu)={\rm{vol}}(\Omega)^{-1}\int_{\Omega}\mathcal{P}_{\vec{k},\nu,\mathcal{I}}(z)e^{-2\pi i{\rm tr}(\mu z)}dz,
\end{equation*}
where for an arbitrary $y\in\mathbb{R}_+^n$, $\Omega=\{z=x+iy: x\in\mathcal{O}_F\setminus\mathbb{R}^n\}$. For our purpose, we choose $y$ with $ N(y)>1$. Now taking the limit as ${k\rightarrow \infty}$ on both sides of the above equation, we obtain
\begin{equation}\label{coe}
 \lim_{k\rightarrow \infty}p_{\vec{k},\nu,\mathcal{I}}(\mu)={\rm{vol}}(\Omega)^{-1}\lim_{k\rightarrow \infty}\int_{\Omega}\mathcal{P}_{\vec{k},\nu,\mathcal{I}}(z)e^{-2\pi i{\rm tr}(\mu z)}dz.
\end{equation}
We want to apply the dominated convergence theorem for interchanging the limit and integration on the right hand side of \eqref{coe}. Now we prove the following two assertions, which enable us to apply the dominated convergence theorem  for the sequence $\{\mathcal{P}_{\vec{k},\nu,\mathcal{I}}\}_{k\geq 1}$.
\begin{enumerate}
\item[{\bf (a)}]
For any $z\in\Omega$, as $k\rightarrow\infty$ we have
\begin{equation}\label{convergence}
\mathcal{P}_{\vec{k},\nu,\mathcal{I}}(z)\rightarrow e{^{2 \pi i{\rm tr}(\nu z)}}.
\end{equation}
 \item[{\bf (b)}]
For all $k >2$ and $z\in \Omega$, there exist an integrable function $G$ on $\Omega$ such that
\begin{equation*}
|\mathcal{P}_{\vec{k},\nu,\mathcal{I}}(z)|\leq G(z).
\end{equation*}
\end{enumerate}

\noindent
To prove the first assertion, we show that as  $k\rightarrow\infty$  exactly one of the term in the series expansion of $\mathcal{P}_{\vec{k},\nu,\mathcal{I}}$ (defined by \eqref{pdefinition}) converges to $e{^{2 \pi i{\rm tr}(\nu z)}}$ and others (individually) tend to $0$. Any term in the series expansion looks like
\begin{equation*}\label{muterm}
\mu(M,z)^{-\vec{k}}e^{2\pi i{\rm tr}(\nu(Mz))},
\end{equation*}
for some $M=\left(\begin{array}{cc}*& * \\ \gamma & \delta\end{array}\right)\in\Gamma_{\infty}\setminus \Gamma_{0}(\mathcal{I})$. Note that $\gamma = 0$ only for one such $M$ and so we can choose $M=I_{\rm id}$ to be a representative. Therefore, in this case
\begin{equation}\label{muequal}
\mu(M,z)^{-\vec{k}}e^{2\pi i{\rm tr}(\nu(Mz))}=e^{2\pi i{\rm tr}(\nu z)},
\end{equation}
for all weight $\vec{k}$ and $z\in \mathbb{H}^n.$ Now suppose $\gamma\neq 0$. For $z=(z_1,z_2,\cdots,z_n)\in \mathbb{H}^n$ it is easy to see that  ${\rm{Im}}({\rm tr}(\nu(Mz)))>0$, which gives
\begin{equation*}\label{mu1}
 |\mu(M,z)^{-\vec{k}}e^{2\pi i{\rm tr}(\nu(Mz))}|\leq |\mu(M,z)^{-\vec{k}}|.
\end{equation*}
Since $0 \neq \gamma \in \mathcal{O}_{F}$, i.e., $N(\gamma)\in \mathbb{Z},$ and hence $N(\gamma)^2\geq 1$. Now
\begin{equation*}\label{mu2}
|\mu(M,z)|^{2}=\prod_{j=1}^{n}|\gamma _jz_j + \delta_j|^2\\
	                                  \geq\prod_{j=1}^{n}(\gamma _j {y}_j)^2\\
                                =N(\gamma)^2N(y)^2\geq N(y)^2,
\end{equation*}
where $y_j={\rm{Im}}(z_j)$ for $1\leq j\leq n.$
Combining the last two inequality, we get
\begin{equation*}\label{muinequality}
|\mu(M,z)^{-\vec{k}}e^{2\pi i{\rm tr}(\nu(Mz))}|\leq N(y)^{-k}.
\end{equation*}
We know that for $z\in\Omega$, $N(y)>1$. Therefore, for any $z\in\Omega$ the above inequality yields
 \begin{equation}\label{muconvergence}
 |\mu(M,z)^{-\vec{k}}e^{2\pi i{\rm tr}(\nu(Mz))}|\rightarrow 0~~\mbox{as}~~k\rightarrow \infty.
 \end{equation}
 Thus the claim \eqref{convergence} follows from \eqref{muequal} and \eqref{muconvergence}.

For the proof of the assertion {\bf (b)}, we define a function $G(z)$ on $\mathbb{H}^n$ by
 \begin{equation*}
G(z)=\sum_{M\in\Gamma_{\infty}\setminus \Gamma_{0}(\mathcal{I})} |\mu(M,z)|^{-3}.
\end{equation*}
It is a well-known (see \cite[lemma 5.7]{fre}) fact that the above series is an absolutely convergent and convergences uniformly in every compact subset of $ \mathbb{H}^n$.

Since the domain $\Omega$ is compact in $\mathbb{H}^n$, therefore $G(z)$ is integrable on $\Omega$, i.e.,
\begin{equation*}
\int_{\Omega}G(z)dz <\infty .
\end{equation*}
Also for any $z\in\Omega$, we obtain
\begin{equation*}
|\mathcal{P}_{\vec{k},\nu,\mathcal{I}}(z)|\leq\sum_{M\in\Gamma_{\infty}\setminus \Gamma_{0}(\mathcal{I})}|\mu(M,z)^{-\vec{k}}e^{2\pi i{\rm tr}(\nu(Mz))}|\leq\sum_{M\in\Gamma_{\infty}\setminus \Gamma_{0}(\mathcal{I})}|\mu(M,z)|^{-3}.
\end{equation*}
In other words, for every positive integer $k>2$ and  $z\in \Omega$, we get
\begin{equation*}\label{bounded}
|\mathcal{P}_{\vec{k},\nu,\mathcal{I}}(z)|\leq G(z),
\end{equation*}
which proves the required result.

Now the dominated convergence theorem allows us to interchange limit and integration in equation \eqref{coe}.
Therefore using \eqref{convergence}, we have
\begin{equation*}
\lim_{k\rightarrow \infty}p_{\vec{k},\nu,\mathcal{I}}(\mu)={\rm vol}(\Omega)^{-1} \int_{\Omega}e^{2\pi i{\rm tr}(\nu z)}e^{-2\pi i{\rm tr}(\mu z)}dz=\delta(\nu,\mu).
\end{equation*}
This completes the proof.
\end{proof}

Next, we prove that the Fourier coefficients $p_{k,\nu,\mathcal{I}}$ also satisfy a similar orthogonality relation with respect to other important parameter, the level $\mathcal{I}$. The proof, as before, uses the dominated convergence theorem for the sequence $\{\mathcal{P}_{k,\nu,\mathcal{I}}(z)\}_{\mathcal{N}(\mathcal{I})\geq 1}.$ Here we work with any weight $k\in\mathbb{N}_{0}^{n}$, not necessarily parallel weight.

\begin{thm}{\bf [Level aspect]}\label{level}
Let $\mathcal{P}_{k,\nu,\mathcal{I}}$ be the $\nu$-${\rm th}$ Hilbert Poincar\'e series of weight $k\in\mathbb{N}_{0}^{n}$ of level $\mathcal{I}$. Then for fixed $k>2,\nu\gg0$ and $\mu\gg 0,$ we have
    $$\lim_{\mathcal{N}(\mathcal{I})\rightarrow \infty}p_{k,\nu,\mathcal{I}}(\mu)= \delta(\nu, \mu).$$
\end{thm}
\begin{proof}
As before, we start with the integral formula
\begin{equation*}
 p_{k,\nu,\mathcal{I}}(\mu)={\rm vol}(\Omega)^{-1}\int_{\Omega}\mathcal{P}_{k,\nu,\mathcal{I}}(z)e^{-2\pi i{\rm tr}(\mu z)}dz.
\end{equation*}
Taking limit $\mathcal{N}(\mathcal{I})$ tends to infinity on the both sides of the above equation, we get
\begin{equation}\label{coe1}
\lim_{\mathcal{N}(\mathcal{I})\rightarrow \infty} p_{k,\nu,\mathcal{I}}(\mu)={\rm vol}(\Omega)^{-1}\lim_{\mathcal{N}(\mathcal{I})\rightarrow \infty}\int_{\Omega}\mathcal{P}_{k,\nu,\mathcal{I}}(z)e^{-2\pi i{\rm tr}(\mu z)}dz.
\end{equation}
For simplifying the right hand side of the above equation, we need to interchange the limit and integration. For that we use the dominated convergence theorem. To apply this theorem, we prove the following two assertions. First, every element in the sequence $\{\mathcal{P}_{k,\nu,\mathcal{I}}(z)\}_{\mathcal{N}(\mathcal{I})\geq 1}$ is bounded by some integrable function on $\Omega$.  Second,
 \begin{equation}\label{convergence1}
\mathcal{P}_{k,\nu,\mathcal{I}}(z)\rightarrow e{^{2 \pi i{\rm tr}(\nu z)}},~~~\mbox{as}~~~ \mathcal{N}(\mathcal{I})\rightarrow\infty,\forall~~ z\in\Omega.
\end{equation}

For $k=(k_1,k_2,\cdots,k_n)\in \mathbb{N}^n$ with $k_j>2$ for all $1\leq j\leq n$, we define
$$G_{k}(z)=\sum_{M\in\Gamma_{\infty}\setminus \Gamma_F}|\mu(M,z)|^{-k}.$$
It is a well-known fact (see \cite[lemma 5.7]{fre}) that the above series is an absolutely convergent and convergences uniformly in every compact subset of $ \mathbb{H}^n$.

Since the domain $\Omega$ is compact in $\mathbb{H}^n$, therefore $G_{k}(z)$ is integrable on $\Omega$, i.e.,
\begin{equation*}
\int_{\Omega}G_{k}(z)dz <\infty .
\end{equation*}
Hence for a fixed $k\in \mathbb{N}^n$ and following the arguments as in the proof of the previous theorem, we have
\begin{equation}\label{bounded1}
|\mathcal{P}_{k,\nu,\mathcal{I}}(z)|\leq G_{k}(z),
\end{equation}
for every integral ideal $\mathcal{I}$ of $\mathcal{O}_{F}$ and for all $z\in \mathbb{H}^n.$ This proves the first assertion.

For the proof of the claim \eqref{convergence1}, we proceed as follows.
Note that $\Gamma_{\infty}\setminus\Gamma_{0}(\mathcal{I})$ is a subset of $\Gamma_{\infty}\setminus\Gamma_{F}$, therefore we write \eqref{pdefinition} as
\begin{equation}\label{pdefinition1}
\mathcal{P}_{k,\nu,\mathcal{I}}(z)=\sum_{M\in\Gamma_{\infty}\setminus\Gamma_{F}}\Delta_{I}(M) \mu(M,z)^{-k}e^{2\pi i{\rm tr}(\nu(Mz))},
\end{equation}
where
\[\Delta_{I}\bigg(\left(\begin{array}{cc}\alpha&\beta\\ \gamma &\delta\end{array}\right)\bigg)=
   \begin{cases}
    1  & ~\text{if}~  \gamma \in \mathcal{I},\\
    0  &~ \text{otherwise}.\\
   \end{cases}
     \]
For $M=\left(\begin{array}{cc}*& * \\ \gamma & \delta\end{array}\right)\in\Gamma_{\infty}\setminus \Gamma_{F}$, consider a general term
\begin{equation}\label{deltaterm}
\Delta_{I}(M) \mu(M,z)^{-k}e^{2\pi i{\rm tr}(\nu(Mz))}
\end{equation}
 in the Poincar\'e series $\mathcal{P}_{k,\nu,\mathcal{I}}(z)$ defined by \eqref{pdefinition1}.
 If $\gamma=0$ then by definition, $$\Delta_{I}(M)=1$$ for all non-zero integral ideal $\mathcal{I}.$ Also $M \in\Gamma_{\infty}$, hence we can take $M$ to be the identity matrix $I_{\rm id}$. Therefore, we have
 \begin{equation}\label{deltaequal}
\Delta_{I}(M) \mu(M,z)^{-k}e^{2\pi i{\rm tr}(\nu(Mz))}=e^{2\pi i{\rm tr}(\nu z)},
\end{equation}
for every non-zero integral ideal $\mathcal{I}.$ Next, if $\gamma\neq 0$, then
\begin{equation*}
\Delta_{I}(M)=0,
\end{equation*}
for all $\mathcal{I}$ with $\mathcal{N}(I)>\mathcal{N}(\gamma\mathcal{O}_F)$.
Hence
\begin{equation}\label{convergence0}
\lim _{\mathcal{N}(I)\rightarrow \infty}\Delta_{I}(M)\mu(M,z)^{-k} e^{2\pi i {\rm tr}(\nu(Mz))}=0.
\end{equation}
The assertion \eqref{convergence1} follows if we take limit $\mathcal{N}(\mathcal{I})$ tends to $\infty$ in the definition
\eqref{pdefinition1} and using \eqref{deltaequal} and \eqref{convergence0}.

Now interchanging the limit and integration in \eqref{coe1} and further using \eqref{convergence1} gives the required result.
\end{proof}

\section{non-vanishing of Hilbert Poincar\'e series}\label{mainresult}
The main goal of the paper is to prove some non-vanishing results for Hilbert Poincar\'e series, which we prove here. Using \thmref{weight} and \thmref{level}, we prove that the Hilbert Poincar\'e series does not vanish identically for sufficiently large weight as well as sufficiently large level respectively.

\begin{thm}\label{nonvanishingw}
Let $\nu\in\mathcal{O}_F^*$, where $\mathcal{O}_F^*$ is the dual space defined by \eqref{lambdaf}. Let $\mathcal{P}_{\vec{k},\nu,\mathcal{I}}$ be the $\nu$-${\rm {th}}$ Hilbert Poincar\'e series of (parallel) weight $\vec{k}$ and level $\mathcal{I}$. Then for fixed $\nu$ and level $\mathcal{I}$, there exist a positive constant $k_0$ such that for all $k>k_0,$ we have
\begin{equation*}
\mathcal{P}_{\vec{k},\nu,\mathcal{I}}\nequiv 0.
\end{equation*}
\end{thm}
\begin{proof}
From \thmref{weight}, we know that for a totally positive $\nu\in\mathcal{O}_F^*$,
\begin{equation*}
 \lim_{k\rightarrow \infty}p_{\vec{k},\nu,\mathcal{I}}(\nu)=1.
\end{equation*}
Hence, there exist some constant $k_0>0$ such that for all $k>k_0$, we have
$$p_{\vec{k},\nu,\mathcal{I}}(\nu)\neq 0,$$
equivalently, $$\mathcal{P}_{\vec{k},\nu,\mathcal{I}}\nequiv 0,$$
which completes the proof of the theorem.
\end{proof}
\begin{rmk}\label{rmkfinite}
More generally, the above theorem is equivalent to the following statement. \linebreak
\em{For any finite set $A \subset \mathcal{O}_F^*$ of indices and for any finite set $B$ of levels, there exist a positive constant $k_0$ such that for all $k>k_0$, we have
\begin{equation*}
\mathcal{P}_{\vec{k},\nu,\mathcal{I}}\nequiv 0,
\end{equation*}
for all $\nu\in A$ and $\mathcal{I}\in B$.}
\end{rmk}

In a particular case of the \thmref{nonvanishingw} when $F=\mathbb{Q}$, we get the following non-vanishing result of Poincar\'e series.
\begin{cor}\label{nonvanishing corollary}
For any positive integer $m$, there exist a positive constant $k_0$ such that for all $k>k_0$,  we have
 \begin{equation*}
\mathcal{P}_{k,m,q}\nequiv 0,
\end{equation*}
where $\mathcal{P}_{k,m,q}$ be the $m$-${\rm{th}}$ Poincar\'e series of weight $k$ and of level $q$.
\end{cor}

Note that \corref{nonvanishing corollary} generalizes an earlier result of Gaigalas \cite{gai} and gives a completely different proof. Moreover, our result is stronger than that.

In the following theorem, we show the non-vanishing of Hilbert Poincar\'e series with respect to level, when the weight is fixed.
\begin{thm}\label{nonvanishingl}
Let $\nu\in\mathcal{O}_F^*$. Then for a fixed $2<k\in \mathbb{N}^n$ and $\nu$, there exist a positive constant $n_0$ such that
\begin{equation*}
\mathcal{P}_{k,\nu,\mathcal{I}}\nequiv 0,
\end{equation*}
for all integral ideal $\mathcal{I}$ with $N(\mathcal{I})\geq n_0$.
\end{thm}
\begin{proof}
A direct application of \thmref{level} and similar arguments as in the proof of \thmref{nonvanishingw}, give the required result.
\end{proof}
\begin{rmk}
Similar to the \rmkref{rmkfinite}, we point out that the above theorem holds good for finitely many weights as well as finitely many indices.
\end{rmk}

Again if the field $F$ is $\mathbb{Q}$ then using \thmref{nonvanishingl}, we have the following non-vanishing result for Poincar\'e series $\mathcal{P}_{k,m,q}$.
\begin{cor}\label{nonvanishing corollary1}
For any positive integer $m$ and for a fixed weight $k\in \mathbb{N}$, there exist a positive constant $q_0$ such that for all $q>q_0$, we have
 \begin{equation*}
\mathcal{P}_{k,m,q}\nequiv 0.
\end{equation*}
\end{cor}

\section{concluding remark}\label{concluding}
It would be interesting to get the analogous result of Rankin, stated in the introduction, for Hilbert Poincar\'e series.

Although our results are weaker than the expected analogous Rankin's result, but the beauty of our proof is that it uses only elementary techniques.

We neither used the explicitly expression for the Fourier coefficients of Hilbert Poincar\'e series nor any estimates for corresponding Bessel function or the generalized Kloosterman sums.

Note that the Fourier coefficients of Siegel Poincar\'e series also satisfy orthogonality relations, proved by Kowalski et. al. \cite{saha}. From this, one can obtained the similar results like \thmref{nonvanishingw} and \thmref{nonvanishingl} for Siegel Poincar\'e series. We also expect that by following the method of this paper, one can prove the analogous non-vanishing results for Poincar\'e series of half-integral weight modular forms.

\begin{acknowledgements}
The author is grateful to her adviser Prof. Brundaban Sahu for his continuous support and useful discussions. She also express her gratitude to Prof. Abhishek Saha for  valuable comments on this paper. The author is greatly indebted to the anonymous referee
for his/her very thorough reading of the manuscript,
also resulting in improved readability of the present version.
\end{acknowledgements}


TITLE: Laurent Series of $\ln(|x|)$?
QUESTION [3 upvotes]: I read in this question that no Laurent series exists for $\ln(x)$, as Patch's answer shows:
"The problem with $\log z$ in the complex plane is that it is a "multi-valued function", so we must specify what range of values we are considering the function to have. Because we must make this choice, the function fails to be continuous in any (punctured) disk about $z=0$, and thus is not (complex) differentiable in any neighborhood about the point."
For me, this raises the question of what if the function were differentiable on a punctured disk about $z=0,\;$  that is, is $\ln(|z|)$ holomorphic, and, if so, what is its Laurent series?
I believe from Owen Biesel's answer here that $\ln(|z|)$ is holomorphic, but I am new to complex analysis, so I could be very wrong. If it is holomorphic,  how would I go about calculating its Laurent series? I understand that it involves taking a positively oriented closed contour integral, but I am having trouble understanding how to properly take contour integrals, as I keep obtaining answers of 0. Thank you in advance!

REPLY [4 votes]: The function $\log |z| = \frac{1}{2} \log (x^2 +y^2)$ is not holomorphic away from $0$, either by the little Picard theorem (the function $\log |e^z|$ would then be real and entire) or because it fails to satisfy the Cauchy-Riemann equations. The linked question asserts that the real part of a holomorphic function is harmonic, which also follows directly from the Cauchy-Riemann equations.


TITLE: Persistent Homology Betti Numbers definition
QUESTION [1 upvotes]: shifting from standard simplicial homology to persistent homology, there is something that I don't understand.
In simplicial homology one builds a chain complex of the form
$$\dots \rightarrow C_n(K) \overset{\partial_n}\rightarrow C_{n-1}(K) \overset{\partial_{n-1}}\rightarrow \dots$$
Where $C_n(K)$ denotes the vector space on the field $\mathbb{Z}_2$ spanned by $n$- dimensional simplices. (i.e. $C_n(K) = \{\sum_{i=1}^p\alpha_i\sigma_i\}$ where each $\sigma_i$ is an $n$-simplex.
Then the $n$-th dimensional homology is defined as $H_n(K) = ker\partial_n(K)/Im\partial_{n+1}(K)$.
Instead, when we shift to persistent homology, what we have is a chain complex for each sub-complex one constructs by varying the filration parameter $\varepsilon$,
$$\emptyset = K^{(0)} \subseteq K^{(1)} \subseteq K^{(2)} \subseteq \dots \subseteq K^{(m)} = K$$
and the filtration is connected by inclusion homomorphism
$$H_n(K^{(0)}) \overset{i_n(0,1)}\longrightarrow H_n(K^{(1)}) \longrightarrow \dots \longrightarrow H_n(K^{(m-1)}) \overset{i_n(m-1,m)}\longrightarrow H_n(K^{(m)}) = H_n(K) $$
for every dimension $n$. At this point, for $i \le j$ the $n$-th persistent homology group is defined as
$$H_n^{(i,j)} := ker\partial_n(K^{(i)})/((Im \partial_{n+1} (K^{(j)}) \cap ker\partial_n (K^{(i)})) $$
I'm just thinking about this modified version of the homology and why do we include the second term in the denominator. If anyone could briedly comment on this and giving some question, it would be highly appreciated.
Many thanks,
James

REPLY [2 votes]: A short answer would be the following, but first, some background! If $H$ is a normal subgroup of a group $G$, then we are able to define a quotient group $G/H$. Since everything in sight is an abelian group, we could instead work with abelian groups: If $H$ is a subgroup of an abelian group $G$, then we are able to define a quotient group $G/H$. Or, since you have chosen field ($\mathbb{Z}/2\mathbb{Z}$) coefficients, we could instead work with vector spaces. If W is a vector subspace of a vector space $V$, then we can define the quotient space $V/W$. Note in all of these cases, the "denominator'' is a suboject of the "numerator".
In
$$H_n^{(i,j)}:=\mathrm{ker}\partial_n(K^{(i)})/(\mathrm{Im}\partial_{n+1}(K^{(j)}) \cap \mathrm{ker}\partial_n(K^{(i)})),$$
if you don't take the intersection in the "denominator", then the "denominator" may not be a subobject of the "numerator"! To be more explicit, $\mathrm{Im}\partial_{n+1}(K^{(j)})$ need not be a subobject of $\mathrm{ker}\partial_n(K^{(i)})$ when $j>i$. That is why one needs to take the intersection. But once you intersect $\mathrm{Im}\partial_{n+1}(K^{(j)})$ with $\mathrm{ker}\partial_n(K^{(i)})$, then certainly the resulting intersection is a subobject of the "numerator" $\mathrm{ker}\partial_n(K^{(i)})$, and hence it is possible to define the quotient.


TITLE: Palindromic Hypotenuses?
QUESTION [2 upvotes]: What is the largest seven-digit palindrome which can be expressed as the sum of two perfect squares?
I tried Java but couldn't get the right answer, and unfortunately OEIS ends at around 5 digits in.

REPLY [2 votes]: It can't end in $99$ since that is $3$ mod $4$.  Therefore it can't start with $99$, so the next-highest prefix is $98$.  Searching numbers of the form $989?989$ quickly yields $9896989$.