I want to update and install macOS Catalina but keep obtaining this error, "(PKDownloadError error 3.)"

Are you using How to upgrade to macOS Catalina?

Information I've read suggests the error is an Internet error, meaning there was a disconnect during the installation. Are you using Wi-Fi or Ethernet? If the former, then try installing with the computer on an Ethernet connection. Other information suggested it is not the Internet. In this case, there have been a couple of suggestions. One is to install from a bootable USB flash drive; the other is to use Recovery Mode. For the latter, there are two approaches. One is to use the Recovery HD installed by macOS; the other is to use Internet Recovery. I'm unsure what to suggest. My first inclination is to try the Recovery HD. It's the easiest one to do. However, I don't know if it will install Catalina or the version you currently have installed. It should do the latter as I understand it. I must alert you that you should have a solid and current backup handy should there be a problem, and because you will need to erase the drive.

Install El Capitan or Later from Scratch

If possible, back up your files.

1. Restart the computer. Immediately, at or before the chime, hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
2. Select Disk Utility from the Utility Menu and click on the Continue button.
3. When Disk Utility loads select the target drive (out-dented entry w/type and size info) from the Device list.
4. Click on the Erase button in Disk Utility's toolbar. A panel will drop down.
5. Set the partition scheme to GUID.
6. Set the Format type to APFS (SSDs only) or Mac OS Extended (Journaled.)
7. Click on the Apply button, then wait for the Done button to activate and click on it.
8. Quit Disk Utility and return to the Utility Menu.
9. Select Install OS X and click on the Continue button.

In Step 3 you will find entry arrangement like below (yours may be different). You want to select the out-dented entry.

Apple SSD SM0512 Media <= out-dented entry

Macintosh HD <= indented entry

When the installation completes successfully, open About This Mac from the Apple menu to find out what version of macOS was installed. If needed you can try the upgrade to Catalina once again. If all is well, then restore your files from your backup.

Now, if the above doesn't help, then you can try doing the Internet Recovery.

Internet/Network Recovery of El Capitan or Later on a Clean Disk

     If possible back up your files before proceeding.

1. Restart the computer. Immediately, at or before the chime, hold down the (Command-Option-R) keys until a globe appears.
2. The Utility Menu will appear in from 5-20 minutes. Be patient.
3. Select Disk Utility and click on the Continue button.
4. When Disk Utility loads select the target drive (out-dented entry w/type and size) from the side list.
5. Click on the Erase button in Disk Utility's toolbar. A panel will drop down.
6. Set the partition scheme to GUID.
7. Set the Format type to APFS (SSDs, only if installing Mojave or Catalina ) or Mac OS Extended, (Journaled.)
8. Click on the Apply button, then click on the Done button when it activates.
9. Quit Disk Utility and return to the Utility Menu.
10. Select Install OS X and click on the Continue button.

Does the error occur while downloading or while installing? If the latter, how far along when it occurs?