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.
E-sharp is called the enharmonic equivalent of F-natural. The sharp raises the base note one half step. In the same way B-sharp is the enharmonic respelling of C-natural. Furthermore, C-flat is the enharmonic of B-natural, and F-flat enharmonic of E-natural. Notes may also be double-sharped and double-flatted, which raises the base note by two half steps.
The key of D-sharp could be considered a theoretical scale because it doesn’t appear on the basic circle of fifths is simply because it is a scale that contains these enharmonics, and traditionally composers write this as E-flat to avoid the enharmonic spelling of the second scale degree, F-natural, which is enharmonic with E-sharp, the second degree of D-sharp major, and no one likes reading parts with double-sharps in the key signature (shudder). The seventh degree of the D-sharp major scale is C-double-sharp. D-sharp major chords definitely exist in music and we play enharmonic spellings all the time.
(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?)
E# is not technically F, only in equal tempered tuning. That's the first cause of concern.
The degrees of any scale follow a sequence and that sequence is never broken. The note names follow the "musical alphabet" {A, B, C, D, E, F, G} then repeat. Accidentals are needed to preserve the whole step - half step sequence {1, 1, 1/2, 1, 1, 1, 1/2} which is two tetra chords separated by a whole step. The numbered degrees are simply {1, 2, 3, 4, 5, 6, 7}. So for example, in the Key of E the degree letter names are {E, F, G, A, B, C, D} repeat. To preserve the steps we raise or lower notes as needed, for this example we get {E, F#, G#, A, B, C#, D#}. The second degree of E is always an F of some sort (period). Now in your example the key is D#. The correct way to name notes is to take D and just make everything #, Key of D = {D, E, F#, G, A, B, C#}. The key of D# would be {D#, E#, F##, G#, A#, B#, C##}. If the E# bothers you, why not ask about the futility of F## and C##? Using enharmonicity of the equal tempered tuning one would usually notate this key as Eb, which takes a lot less effort to notate correctly.
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.
D♯ E♯ F𝄪 G♯ A♯ B♯ C𝄪 D♯?