Hi the problem is in FileDownloader class

 urlConnection.setRequestMethod("GET");
    urlConnection.setDoOutput(true);

You need to remove the above two lines and everything will work fine. Please mark the question as answered if it is working as expected.

Latest solution for the same problem is updated Android PDF Write / Read using Android 9 (API level 28)

Attaching the working code with screenshots.

MainActivity.java

package com.example.downloadread;

import java.io.File;
import java.io.IOException;

import android.app.Activity;
import android.content.ActivityNotFoundException;
import android.content.Intent;
import android.net.Uri;
import android.os.AsyncTask;
import android.os.Bundle;
import android.os.Environment;
import android.view.Menu;
import android.view.View;
import android.widget.Toast;

public class MainActivity extends Activity {

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);
    }

    @Override
    public boolean onCreateOptionsMenu(Menu menu) {
        // Inflate the menu; this adds items to the action bar if it is present.
        getMenuInflater().inflate(R.menu.main, menu);
        return true;
    }

    public void download(View v)
    {
        new DownloadFile().execute("http://maven.apache.org/maven-1.x/maven.pdf", "maven.pdf"); 
    }

    public void view(View v)
    {
        File pdfFile = new File(Environment.getExternalStorageDirectory() + "/testthreepdf/" + "maven.pdf");  // -> filename = maven.pdf
        Uri path = Uri.fromFile(pdfFile);
        Intent pdfIntent = new Intent(Intent.ACTION_VIEW);
        pdfIntent.setDataAndType(path, "application/pdf");
        pdfIntent.setFlags(Intent.FLAG_ACTIVITY_CLEAR_TOP);

        try{
            startActivity(pdfIntent);
        }catch(ActivityNotFoundException e){
            Toast.makeText(MainActivity.this, "No Application available to view PDF", Toast.LENGTH_SHORT).show();
        }
    }

    private class DownloadFile extends AsyncTask<String, Void, Void>{

        @Override
        protected Void doInBackground(String... strings) {
            String fileUrl = strings[0];   // -> http://maven.apache.org/maven-1.x/maven.pdf
            String fileName = strings[1];  // -> maven.pdf
            String extStorageDirectory = Environment.getExternalStorageDirectory().toString();
            File folder = new File(extStorageDirectory, "testthreepdf");
            folder.mkdir();

            File pdfFile = new File(folder, fileName);

            try{
                pdfFile.createNewFile();
            }catch (IOException e){
                e.printStackTrace();
            }
            FileDownloader.downloadFile(fileUrl, pdfFile);
            return null;
        }
    }


}

FileDownloader.java

package com.example.downloadread;

import java.io.File;
import java.io.FileNotFoundException;
import java.io.FileOutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.net.HttpURLConnection;
import java.net.MalformedURLException;
import java.net.URL;

public class FileDownloader {
    private static final int  MEGABYTE = 1024 * 1024;

    public static void downloadFile(String fileUrl, File directory){
        try {

            URL url = new URL(fileUrl);
            HttpURLConnection urlConnection = (HttpURLConnection)url.openConnection();
            //urlConnection.setRequestMethod("GET");
            //urlConnection.setDoOutput(true);
            urlConnection.connect();

            InputStream inputStream = urlConnection.getInputStream();
            FileOutputStream fileOutputStream = new FileOutputStream(directory);
            int totalSize = urlConnection.getContentLength();

            byte[] buffer = new byte[MEGABYTE];
            int bufferLength = 0;
            while((bufferLength = inputStream.read(buffer))>0 ){
                fileOutputStream.write(buffer, 0, bufferLength);
            }
            fileOutputStream.close();
        } catch (FileNotFoundException e) {
            e.printStackTrace();
        } catch (MalformedURLException e) {
            e.printStackTrace();
        } catch (IOException e) {
            e.printStackTrace();
        }
    }
}

AndroidManifest.xml

<?xml version="1.0" encoding="utf-8"?>
<manifest xmlns:android="http://schemas.android.com/apk/res/android"
    package="com.example.downloadread"
    android:versionCode="1"
    android:versionName="1.0" >

    <uses-sdk
        android:minSdkVersion="14"
        android:targetSdkVersion="18" />
    <uses-permission android:name="android.permission.INTERNET"></uses-permission>
    <uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE"></uses-permission>
    <uses-permission android:name="android.permission.ACCESS_NETWORK_STATE"></uses-permission>
    <uses-permission android:name="android.permission.READ_PHONE_STATE"></uses-permission>
    <application
        android:allowBackup="true"
        android:icon="@drawable/ic_launcher"
        android:label="@string/app_name"
        android:theme="@style/AppTheme" >
        <activity
            android:name="com.example.downloadread.MainActivity"
            android:label="@string/app_name" >
            <intent-filter>
                <action android:name="android.intent.action.MAIN" />

                <category android:name="android.intent.category.LAUNCHER" />
            </intent-filter>
        </activity>
    </application>

</manifest>

activity_main.xml

<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:tools="http://schemas.android.com/tools"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:paddingBottom="@dimen/activity_vertical_margin"
    android:paddingLeft="@dimen/activity_horizontal_margin"
    android:paddingRight="@dimen/activity_horizontal_margin"
    android:paddingTop="@dimen/activity_vertical_margin"
    tools:context=".MainActivity" >

    <Button
        android:id="@+id/button1"
        android:layout_width="match_parent"
        android:layout_height="wrap_content"
        android:layout_alignParentLeft="true"
        android:layout_alignParentTop="true"
        android:layout_marginTop="15dp"
        android:text="download"
        android:onClick="download" />

    <Button
        android:id="@+id/button2"
        android:layout_width="match_parent"
        android:layout_height="wrap_content"
        android:layout_alignParentLeft="true"
        android:layout_alignParentRight="true"
        android:layout_below="@+id/button1"
        android:layout_marginTop="38dp"
        android:text="view"
        android:onClick="view" />

</RelativeLayout>

You said in the comments the PDF files have to be public to the user (in the file explorer or in a PDF reader for example) so I updated my answer.

Before Android R from 2020 it was easy for an app to create files in the (public) external storage directory. "external" only means that the directory is not an internal app directory. It doesn't mean that the directory is part of an external SD card for example. Files in this directory (and child directories except the "Android" folder) are shown by the gallery app for example or by PDF readers.

In my Android API level 33 emulator the path to the (public) external storage directory is /storage/emulated/0. The directory also contains some default child directories named "Documents", "Download" or "Pictures" for example. Because the path is not guaranteed to always be the same you retrieve it using getExternalStorageDirectory or getExternalStoragePublicDirectory for one of the child directories.

So before Android R all you had to do is request permission WRITE_EXTERNAL_STORAGE and put your file into the directory returned by getExternalStoragePublicDirectory(DIRECTORY_DOCUMENTS).

Unfortunately Android gets more strict every release and now WRITE_EXTERNAL_STORAGE has no effect any more and getExternalStoragePublicDirectory is also deprecated.

So to put your file into the (public) external Documents directory on Android R and later there seem to be multiple ways.

One way is to declare permission MANAGE_EXTERNAL_STORAGE in the manifest:

<uses-permission android:name="android.permission.MANAGE_EXTERNAL_STORAGE"/>

Then ask the user to enable it manually:

Uri u = Uri.parse("package:" + BuildConfig.APPLICATION_ID);
Intent intent = new Intent(Settings.ACTION_MANAGE_APP_ALL_FILES_ACCESS_PERMISSION, u);
startActivity(intent);

After this you can write to the folder returned by getExternalStoragePublicDirectory(DIRECTORY_DOCUMENTS) which points to /storage/emulated/0/Documents on my emulator.

There are two more ways described in this answer. One way is to use ACTION_CREATE_DOCUMENT and the other is to use MediaStore.

I also want to say that after placing your file in the (public) external storage directory it may take some time until it appears in PDF readers. You can send intent ACTION_MEDIA_SCANNER_SCAN_FILE so this happens faster. See this answer.

I found a package to solve this case.
This way is very easy and working fine.
So, I set this package as the best solution for now.
If anyone have a better way, I will define it as answer.

Like this:

Intent browserIntent = new Intent(Intent.ACTION_VIEW, Uri.parse("*url for your pdf*"));
startActivity(browserIntent);

This will cause the pdf to be downloaded in the notification bar, as if you had initiated the download through the browser.

Check this question if you have problems: How can I open a URL in Android's web browser from my application?

You can view or download the pdf by either of the two ways i.e by opening it in device in-built browser or in the webview by embedding it in your app.

To open the pdf in browser,

Intent browserIntent = new Intent(Intent.ACTION_VIEW, Uri.parse(pdf_url));
startActivity(browserIntent);

Instead to open in webview,

 WebView webView = (WebView) findViewById(R.id.webView1);
 webView.getSettings().setJavaScriptEnabled(true);
 webView.loadUrl(pdf_url);

Download a pdf:

 startActivity(new Intent(Intent.ACTION_VIEW, Uri.parse("www.education.gov.yk.ca/pdf/pdf-test.pdf")));

Ah as I've just found out this is device dependent.

scenarios

1. It will download the pdf to your browser/downloaded/ folder

2. You have a google docs account - It will ask you to sign in then view the pdf in the browser

3. You have a pdf reader installed - App dependent may catch it may not

In all scenarios though the user has access to the PDF with one line of code :-)