Question: I've been having this doubt for a quite a while and haven't been able to find a satisfying answer. Is the 'E#' note technically an F? If so, why does it not exist on the chromatic scale like other sharps? I mean, what exactly is the E#? Is it a note on its own or is it just another way of saying F?

And what do people mean when they say that the D# Major scale is theoretical major scale and why it does not exist in the circle of fifths diagram?

I came across the last paragraph on a website.

Answer: 
(In addition to Tim's great answer, I wanted to introduce another important term and a slightly different way of thinking of it.)

Whenever we see a pitch that is raised from its expected form in the diatonic scale, it is often (although not always) understood as a *temporary leading tone*. By "leading tone" we mean it's the seventh scale degree, and since E♯ is the seventh scale degree of an F♯ scale, we say that this E♯ is the temporary leading tone of F♯.

Even though this E♯ is the exact same note as F on a modern equally tempered piano, their spellings tell us how the two pitches function. E♯ is scale-degree 7 of F♯, whereas F is scale-degree 7 of G♭. Even though E♯ and F sound the same—we say that they are \*enharmonic—they have very different functions.

As for D♯ major being a theoretical key, this just means that it has at least one doublesharp or doubleflat in its key signature. D♯ actually has *two* doublesharps—F and C—so it is a theoretical key. (See also [Where do the double accidentals go in "theoretical" key signatures?](https://music.stackexchange.com/questions/65936/where-do-the-double-accidentals-go-in-theoretical-key-signatures))
For starters, there isn't an E# in the D major scale. There's E and there's F#. If you're asking why there may be E# found in a piece that's in D major, that's slightly different.

Sometimes certain notes that are non-diatonic need to be used. That's maybe where E# could come in. Let's say the chord sequence goes from D major into F# minor. There's often the V of the new chord that precedes it. Here, it would be C# or C#7. Now the spelling for that is C# E# and G#. It is **not** spelled C# F and G#.

There are other times and reasons why some notes are written in 'strange' ways, but there are generally good technical reasons for them. And just because these notes happen to be white keys on a piano doesn't mean they can't be called # or b. These being - B#, Cb, E# and Fb. Of course, for beginners, things like E# sound pretty silly, why not F instead? Correctness and technical reasons!

**EDIT** - now the question has been edited, this answers a rather different one! Thanks! I'll leave this up for now, (OP originally asked about E# in a D major key), as it goes a little way. Luckily, I noticed the edit, otherwise I'd look even more stupid than usual.