If you have servers in different time zomes, or with funny notions of time which somehow work out all right, you could call gmtime on the $modtime for accurate comparison. gmtime acts just like localtime, but produces UTC time.

use POSIX 'strftime';
for (glob '*') {
    printf "%s\t%s\n",
            $_,
            strftime('%B %d', gmtime((stat)[9]));
}
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In list context, gmtime and localtime both produce properly normalized time structs from epoch time.