theme_update_report in core/modules/update/update.theme.inc

Returns HTML for the project status report.

update_calculate_project_update_status in core/modules/update/update.compare.inc

Calculates the current update status of a specific project.

update_get_available in core/modules/update/update.module

Tries to get update information from cache and refreshes it when necessary.

_update_get_cached_available_releases in core/modules/update/update.module

Returns all currently cached data about available releases for all projects.

_update_message_text in core/modules/update/update.module

Returns the appropriate message text when site is out of date or not secure.

_update_requirement_check in core/modules/update/update.install

Fills in the requirements array.