As I understand the documentation, the variable
'display-time-mail-function' should allow to provide a function which
decides whether to display the mail icon in the status area.

Unfortunately the code does not allow this semantic, as the code uses an
or expression over several possibilities (lisp/time.el:407):

        (mail (or (and display-time-mail-function
                       (funcall display-time-mail-function))
                  (and display-time-mail-directory
                       (display-time-mail-check-directory))
                  (and (stringp mail-spool-file)
                       (or (null display-time-server-down-time)
                           ;; If have been down for 20 min, try again.
                           (> (- (nth 1 now) display-time-server-down-time)
                              1200)
                           (and (< (nth 1 now) display-time-server-down-time)
                                (> (- (nth 1 now)
                                      display-time-server-down-time)
                                   -64336)))
                       (let ((start-time (current-time)))
                         (prog1
                             (display-time-file-nonempty-p mail-spool-file)
                           (if (> (- (nth 1 (current-time))
                                     (nth 1 start-time))
                                  20)
                               ;; Record that mail file is not accessible.
                               (setq display-time-server-down-time
                                     (nth 1 (current-time)))
                             ;; Record that mail file is accessible.
                             (setq display-time-server-down-time nil)))))))


So when display-time-mail-function is set but returns nil, then the
following tests will still be evaluated instead of using this negative
result.

As a fix I propose the attached patch.

Thanks
  Detlev


In GNU Emacs 24.0.50.1 (i686-pc-linux-gnu, GTK+ Version 2.18.9)
 of 2010-08-16 on ohwell
Windowing system distributor `The X.Org Foundation', version 11.0.10707000


-- 
... and I will continue to use a  low-level language to indicate how
machines actually compute.  Readers  who only want to see algorithms
that are already packaged in a plug-in way, using a trendy language,
should buy other people's books.
                                         -- Donald Knuth, Fascicle 1