This is normal behavior of the mouse.
When the mouse is first turned on or you replace the batteries, the green battery light will come on and stay on for a few seconds as shown below:

The green light then goes off to conserve battery power.
Important Notice: The Firmware Update Tool is no longer supported or maintained by Logitech. We highly recommend using Logi Options+ for your supported Logitech devices. We are here to assist you during this transition.
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Important Notice: The Logitech Control Center is no longer supported or maintained by Logitech. We highly recommend using Logi Options+ for your supported Logitech devices. We are here to assist you during this transition.
Important Notice: The Logitech Connection Utility is no longer supported or maintained by Logitech. We highly recommend using Logi Options+ for your supported Logitech devices. We are here to assist you during this transition.
Important Notice: The Unifying Software is no longer supported or maintained by Logitech. We highly recommend using Logi Options+ for your supported Logitech devices. We are here to assist you during this transition.
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