If you are seeing a blank page when trying to update your WordPress installation it is probably because you are using a caching plugin. Try switching the plugin off before commencing with the upgrade.
This worked for me today when update-core.php was returning a blank white page whilst trying to upgrade this blog to WordPress 3.6. I found a hint that pointed to my issue here: http://core.trac.wordpress.org/ticket/22270
I deactivated W3 Total Cache, performed an upgrade then turned it back on afterwards without issue.
Blank page when updating WordPress.