Available online here:

http://www.rickenbacker.com/pdfs/manual.pdf

I may be rather new here in this forum, but I went to nearly all threads dealing with the "E-string" topic.
I know this thread is old - nonetheless it's helpful.

As I bought a brand new 4003 recently and could figure out somehow what you were talking about - I have completely to agree with Wim - there is a lot of output on the neck PU with the E-string but you probably wouldn't notice because many players are not used to hear the fundamental of an "E" but its next harmonics. The so-called 'solid underlying bass' is exactly that what you can hear/feel if only the rig is capable to make it hearable (or give enough pressure onto the speakers).
I have been playing 5-string basses for more than 2 decades and had to adapt the rig accordingly...

Well, you may miss some of the next harmonics that we can make out clearly on the A or D string, but imho this is a fundamental constructional part of the whole concept of 4001/4003 basses.
The position of the neck PU in relation to the effective string length gives an essential root cause for suppression and/or intensity of certain frequencies.
It's just about wave physics, spread of material properties included. No piece of wood is identical to another one.

Please don't get me wrong - I highly appreciate the great work on these unique instruments and I would love my 4003 forever :)

It might be that your amp can't reproduce the sound of the E correctly.

Exactly. I was just in the studio, where the monitors were well equipped to produce very low frequencies audibly (much unlike many bass cabinets!!). I only used my neck pickup to record all of the tracks and the bass was as loud and "in the mix" as ever; yet I have often thought that pickup was quiet when running it through a bass cab. I'd like a bass cab that could produce lows as well as a studio monitor.