Your friend is correct. From a strict "gold per stat" cost analysis, Phantom Dancer's upgrade cost gives more bang for the buck than Zeal. I break down each stat on its own merit, but you have to take the composite (which is even more in PD's favor) to get the full picture.
Zeal provides 20% Attack Speed. Phantom Dancer provides 55%, meaning that the upgrade over Zeal is 35% Attack Speed for the cost of the item. The price per stat then looks like:
- Zeal: 1195 Gold / 20% ASPD = 59.75 Gold per Percent ASPD
- Phantom Dancer Upgrade: 1650 Gold / 35% ASPD = 47.14 Gold per Percent ASPD
It is worth noting that two Zeals still gives the same Gold to Percent ASPD ratio. ASPD is also not multiplicative as far as items go, meaning that you do not get a higher final Attack Speed from two Zeals due to two multiplications by 1.20 instead of one by 1.55. (1.20 squared is still lower than 1.55, regardless.)
Critical Strike Chance
Zeal provides 10% Crit Chance. Phantom Dancer provides 30% - this makes Phantom Dancer an upgrade of 20% Crit Chance for the cost. This gives us a price per stat comparison of:
- Zeal: 1195 Gold / 10% Crit Chance = 119.50 Gold per Percent Crit
- Phantom Dancer Upgrade: 1650 Gold / 20% Crit Chance = 82.50 Gold per Percent Crit
Again, much like ASPD, two Zeals has the same price per stat point, and still doesn't compare to a Phantom Dancer. Crit chance is also additive.
Movement speed is where Zeal earns a slight edge over the Phantom Dancer. Zeal's MS bonus is 6%; Phantom Dancer's is 12%, meaning the upgrade is merely another 6%. Price per point thus looks like:
- Zeal: 1195 Gold / 6% MS = 199.16~ Gold per Percent Movespeed
- Phantom Dancer Upgrade: 1650 Gold / 6% MS = 275 Gold per Percent Movespeed
It is important to note that, when considering Movement Speed, there are numerous soft caps/diminishing returns plateaus. Detailed on League Wiki, the gist is that any movespeed above 415 is reduced by 20%, and any movespeed above 490 is reduced by 50% (not cumulative). With Berserker Greaves and Zeal, any champion with base 320 or 330 movement speed is bumping into that 415 cap - both a second Zeal and their Phantom Dancer would be hit with the reduction. Most AD carries tend to be between 300 and 310 base MS, however, but quite a few have things that boost that, like Vayne, Sivir, or any carry that is running with Nunu support.
With the exception of Movement Speed - which in quite a few cases is already decayed by diminishing returns and is not really the hallmark stat of Zeal or Phantom Dancer, it is (from a gold perspective) almost always a better buy to upgrade Zeal into PD, as opposed to purchasing a second. Raven Dreamer's point about item slot efficiency is also important to consider - item space is often more precious than gold to an AD carry, due to the number of tools they need to succeed (Berserker Greaves, Infinity Edge, Last Whisper, some sort of Lifesteal item, and some sort of defensive item). The popularity of multiple early-game Doran's Blades for early game durability merely compounds this issue, and makes the advantage even more favorable for the singular, more cost-efficient item - the Phantom Dancer.