Arsenal have made three (maybe four) first-team signings this summer.
They have replaced the gaping hole at center forward with Gabriel Jesus and promoted Eddie Nketiah to be the primary backup.
They have replaced the probably very soon-to-be leaving Bernd Leno with Matt Turner, giving Arsenal a solid backup goalkeeper.
They have also added some creative depth with Fabio Vieira. Interestingly I don't think that this really is a direct replacement for a player in the squad but rather cover for an area that the team took a risk on last season.
That leads us to the next question of what areas of the team still seem to be upgraded. A good place to start I think is looking at the team's depth chart and getting a feel for the vibes they are giving off.
Goalkeeper - Without a shock massive injury or something like that I think this position is well covered. No more business here.
Right back - Arsenal have an established starter here and then have Cedric plus the returning Hector Bellerin. I think that it is highly likely that Bellerin leaves and a very low probability that Cedric leaves. For any moves in this position, I think it would take both backups leaving. Very low priority.
Center back - Arsenal have a starting pair, a solid backup in Rob Holding plus the returning William Saliba and Pablo Mari. That is plenty cover for the matches Arsenal have but I think that there is certainly room to replace Mari, especially if that player has the ability to cover other positions on the left with a player like Lisandro Martinez. Low priority.
Left back - I have already gone on the record saying that I would be okay with selling Kieran Tierney but I also think that he is perfectly fine as the starting left back. Where Arsenal might need some help is still with the player behind him, this is a spot where the backup is more important than others given the injury track record of Tierney. I think Nuno Tavares is an exciting but raw player and I can understand not having the full trust of the coach to play 10 or more starts. With Martinez rumors, I think that this makes a lot of sense given that Arsenal see him as a left back (with the ability to play in other spots as well). Medium Priority.
Midfield - This one just got potentially complicated with recent news and lots of speculation about who this player might be.
As an American, it is a bit jarring that arrest information is under a gag order where all that is left is vague descriptions and people trying to make connections from that. I guess we will have to wait and see if a charge is actually made here for the player to be officially named.
Even without this potential complication, I think that this was an area of need for Arsenal. Not necessarily that Arsenal don't have the numbers but as an area where a new starting-caliber player could make a large impact in the first eleven. The obvious area is in the left eight for a player that is a more natural fit for the role compared to Granit Xhaka. Very High Priority.
Attacking Midfield - Arsenal have five nailed-on players for these three starting spots, plus Nicolas Pepe looks like he is on his way out. If Pepe leaves I think he is a player that must be replaced, having just five players that you trust in those spots for the 44+ very high leverage matches Arsenal will expect to play this year is just not enough coverage. Getting that to six players is important and it makes a ton of sense why Arsenal have been knee deep in the Raphinha sweepstakes. High Priority
Striker - Arsenal have two guys here and I wouldn't be surprised if Florian Balogun sticks around for at least the first half of the season to make it three players to give some cover in the early cups and Europa League group stages. No more business here.
Looking at my list, I have midfield as the top priority. It is a little concerning that this seems to have gone a bit quite because I was pretty certain that there was a very good shot of Youri Tielemans joining the team in time for a full pre-season. Hopefully, this is just something that Arsenal feel like there isn't a lot of outside interest and they don't want to be bidding against themselves in an area where Leicester don't have a ton of leverage.
Next, I have attacking midfield and I think that this is one where the business is more public because the selling team really wants to spark a bidding war. If Arsenal do end up missing out on Raphinha I think that this is an area that the team will revisit but might be one that waits for closer to deadline day to see what dominos fall to loosen up other players.
Last I have the left back area. I think Arsenal have identified Martinez because yes he can play left back but also would give cover as the left footed center back and potentially deep cover for midfield. If Arsenal miss out here it might be another one that waits for the end of the window.