BibleHub.com has an interlinear translation (word-by-word translation of Greek to English) of this verse. The Greek reads:
ἦσαν δὲ προσκαρτεροῦντες τῇ διδαχῇ τῶν ἀποστόλων, καὶ τῇ κοινωνίᾳ, τῇ κλάσει τοῦ ἄρτου, καὶ ταῖς προσευχαῖς.
The word by word English translation reads
"they were moreover steadfastly continuing in the teaching of the apostles and - in fellowship the breaking of the bread and the prayers.
The word "the" in "the prayers" is the standard Greek article (in the appropriate gender, case, and number), ταῖς. One would think that the phrase should be translated "in the prayers", as the ESV has it.
On the other hand, the dash between and and in fellowship corresponds to a similar article that is not translated. A word-by-word translation of that phrase, καὶ τῇ κοινωνίᾳ, would be "and in the fellowship". Yet most translations (the ESV seems to be an exception) make that phrase "in fellowship". And indeed "in the fellowship" sounds a bit awkward to me.
The root of the problem is simply that Greek puts in definite articles before words in situations where English doesn't, and even the ESV seems to remove them sometimes. Acts 2:40, for example, contains the Greek phrase
Σώθητε ἀπὸ τῆς γενεᾶς τῆς σκολιᾶς ταύτης
A literal, word-by-word translation is
Be saved from the generation the perverse this
The ESV removes both articles in its translation,
Save yourselves from this crooked generation
Thus it appears that the phrasing "the prayers" in Acts 2:42 is simply a choice by the translators to translate the Greek literally in this instance, rather than a reference by the author to specific prayers.