How to query users who have empty first_name?

Question

It looks like you’re searching for the first_name meta keys with empty string meta values:

$options = array(
    'meta_query' => array(
        array(
            'key'     => 'first_name',
            'value'   => '',
            'compare' => '=',
        ),
    )
);
$users = get_users( $options );

that generates this kind of SQL query:

SELECT wp_users.* 
FROM wp_users 
INNER JOIN wp_usermeta ON ( wp_users.ID = wp_usermeta.user_id ) 
WHERE 
    1=1 AND ( ( wp_usermeta.meta_key = 'first_name' AND wp_usermeta.meta_value="" ) ) 
ORDER BY user_login ASC

Note that

$options = array(
    'meta_key'     => 'first_name',
    'meta_value'   => '',
    'meta_compare' => '=',
);
$users = get_users( $options );

generates this kind of SQL query:

SELECT wp_users.* 
FROM wp_users 
INNER JOIN wp_usermeta ON ( wp_users.ID = wp_usermeta.user_id ) 
WHERE 1=1 AND ( wp_usermeta.meta_key = 'first_name' ) 
ORDER BY user_login ASC ;