I decided to install the Logitech Gaming Software for my Logitech g203. I went to the support page for the mouse, and downloaded the software it suggested. When I tried to install the software it had the error "Sorry, this software is not compatible with your operating system. Please visit http://www.logitech.com/support to download a compatible version."
I'm using Windows 10 64-bit, and for some unknown reason the logitech website was automatically detecting that my operating system is 32-bit (x86) and giving me the 32-bit version of the software. By clicking the "Show All Downloads" button I was able to quickly find the 64-bit version, which does work. The download listed under "Show All Downloads" was also a newer version of the software. Why the Logitech website is suggesting that users download an older version that's not compatible with their operating system, I cannot guess.