Leo also has an example using "stating" to translate "unter Angabe", so this could simply be deleted.

I don't believe this is in any way standard English: it sounds like a direct word-for-word mistranslation from the German. I found it when proofreading a text written by a German and trying to work out what they meant by "under specification of" - I had no idea whatsoever and it was only when I found "unter Angabe von" that the sentence made any sense to me.

I've looked through a few pages of Google UK results to see if people do somehow use this in a specialised context, but all I can see is German translations of "unter Angabe von" or people writing about the underspecification of various things.

Collocation dictionaries do not have "under" as a possible collocation for "specification".

"Specification" is not listed in the dictionary as being a mass noun and thus not requiring an article.

My bilingual dictionaries do not contain "unter Angabe", but "ohne Angabe" is always translated as "without ...ing", not with the noun "specification".

If anything, using "specify", you might say "specifying" - as in "Send in an application, specifying the reasons why you think you are suitable ..."