What did Gandalf look for in archives of Minas Tirith?

[JeyEn]

In Unfinished tales, it is said that

Little is known of Gandalf's history until the end of the Watchful Peace (2460) and the formation of the White Council (2463), and his special interest in Gondor seems only to have been shown after Bilbo's finding of the Ring (2941) and Sauron's open return to Mordor (2951). His attention was then (as was Saruman's) concentrated on the Ring of Isildur; but in his reading in the archives of Minas Tirith he may be assumed to have learned much about the palantíri of Gondor, though with less immediate appreciation of their possible significance than that shown by Saruman, whose mind was in contrast to Gandalf's always more attracted by artefacts and instruments of power than by persons

A footnote to this says:

Note the passage in The Two Towers IV 5 where Faramir (who was born in 2983) recollected seeing Gandalf in Minas Tirith when he was a child, and again two or three times later; and said that it was interest in records that brought him. The last time would have been in 3017, when Gandalf found the scroll of Isildur. [Author's note.]

So in my previous discussion, I was wondering about Gandalf's knowledge about the One. This quotes above suggests Gandalf went to archives of Minas Tirith after Bilbo found the Ring. Therefore, it probably means Gandalf suspected Bilbo found the One and went to archives to look for some proofs?

Or did he just want to find some stuff about Rings of Power, which he didn't know much about back then? But why - he could have gone to Rivendell for something (ask Elrond?) But why - Rings of Power and The One were discussed in the White Council many times.

But my main question is - what did he look for then? It was long before:

1. Bilbo's birthday party and his weird behaviour.
2. Gandalf remembered Saruman's words about the fiery words on the One.
3. Gandalf started to suspect that Bilbo found the One.
4. Gandalf even had a need to look for some stuff about Bilbo's ring.

Something in this doesn't make sense to me, I am missing something. What do you guys think?