It wouldn't improve the standby time at all since 
    that's determined by the radio, not the screen.

In smartphones the display is the biggest consumer of battery.

For instance, I rarely use my Android smartphone, other than for making calls / reading emails (and I really don't do that frequently), and the display is usually eating up ~ 30% of my battery, even though I always keep my data connection + sync ON. And there's nothing that drains my battery faster than doing Internet browsing, with the stats on my Android showing the display taking more than 50% of battery life. And if I'm outside, in the sun, browsing the web ; to make the screen readable I have to increase luminosity, and so my phone drops dead in 2-3 hours.

Another example - I also have an old Nokia 1100 - it's absolutely insane how the battery lasts for a whole week. If you can make your smartphone last for 6-7 days with whatever you want turned off, let me know ;)

I would also like an e-Ink smartphone, since it would be perfect for reading emails, in direct sunlight and also increase the battery life to more than 1 day - yeah, that would be awesome.

If the screen uses 30% of your battery then switching to e-ink would be a 30% improvement at best. Hardly Nokia 1100 territory. Though it would be so slow and annoying to use that you wouldn't use it as much, so that would "help" too. Also, need I remind you that your Nokia 1100 has an LCD screen?

The Nokia 1100 has an LCD screen indeed, but it is black and white, without much luminosity.

Right; the only resl advantage e-ink has over this type of low-power LCD is brightness, and it comes at the cost of 1+ second response time with flickering. A terrible trade off for an interactive phone.

Looking at battery use statistics in my phone:

Voice calls - 33% Android System - 13% Maps - 12% Display - 11% Dialer - 11% Youtube - 10% Cell standby - 7% Phone idle - 4% Camera - 3%

The display does not seem to be particularly bad.

I saw a grey-scale style calendar (branded with a pharmaceutical brand) in a pharmacy the other day. I asked if it was an e-ink display and they said yes. Then they realised it was just an old-style flip-page calendar which LOOKED like e-ink.