"Just build armband against Zed" A mathematical + economic analysis of Armband's effectiveness
This is one of the worst arguments I've seen on the League Boards. While it comes from a mathematically sound argument, it also comes from a logically invalid argument. It assumes that a mage's primary goal is to avoid death from Zed, rather than to win the game. I will not be arguing whether Zed is OP or not, because that's not relevant.
Why is Seeker's Armguard not a viable counter to Zed early game for mana-based AP champions?
Before I started analysis, I roughly estimated that once I get Seeker's Armguard, I'll only get maybe 10% bonus damage reduction against Zed. I did analysis later, and my suspicions were confirmed.
Seeker's Armguard:
http://leagueoflegends.wikia.com/wiki/Seeker%27s_Armguard
Seeker's Armguard grants 30 armor and 20 AP initially, requiring you to cs 30 times in order to reach its max of 45 armor and 35 AP. To a Zed, this looks daunting. 45 whole armor? The equation for how much armor (and MR) resist is calculated with this fuction:
100 / (100 + Resistance value)
At a flat rate, this means that getting 45 armor will mean you only deal 10/14.5, or 69% of the damage you would be dealing, which seems like a huge deal.
However, base armor values are in play as well. Let us take the stance of Cassiopeia, one of the sturdier armor resistant AP midlane mages (outshined by Katarina and Kayle), with a base armor of 25 [+3.5 (level - 1)] armor. Assume both are at Level 5, effectively removing the ultimates from the argument for the sake of early game arguments. Cassiopeia will have 25 (+ 3.5 x 4) armor, meaning she has 39 armor. This means that Zed deals about 69% physical damage to her with no items.
Assuming that Cass finishes a wave and gets 6 cs before deciding to engage Zed, she'll have 33 armor from Seeker's Armguard.
100 / (100 + 39 + 33) = 100/172, meaning Zed will deal 58% physical damage against Cassiopeia. However, Cassiopeia naturally has 69% damage reduction from physical sources, so really it's only a 11% damage decrease.
Now Cass has fully stacked them, and reached level 6. She now has magically masterfully farmed all 30 required cs for the armband, and now has 15.5 additional armor from the last scenario.
100 / (100 + 39 + 33 + 15.5) = 100/187.5, which is about 53%. Seeker's Armguard provides an additional 16% damage negation from Seeker's Armband, which really is quite a lot.
A typical Zed path will have something else that is 1100 gold as well. Serrated dirk. With 25 attack damage, 10 lethality, and Headhunter, Zed also gains power. For the sake of this post, I will ignore the other two components, even if they are significant, and focus on lethality.
Flat Armor Penetration = LETHALITY × (0.6 + 0.4 × level ÷ 18), so Zed will have 7.3 flat armor penetration.
100 / (100 + 39 + 33 + 15.5 - 7.3) = 100/ 180.2, which is 55.4%. Lethality really only increases damage by 2.4%. Lethality is more effective the LOWER a target's armor is, so stacking armor counters lethality.
Keep in mind that I have used Cassiopeia, the highest armor mana mage that is commonly played midlane. This means that both seekers' AND lethality will be proportionally more powerful the lower the armor a mage has, making items seem more effective than they may be in other matchups.
Seekers costs 1100 gold, but removes an essential item for many mana mages now, which is Lost Chapter. Or, tear-stacking mages are unable to prioritize building their main path, and must wait before getting Seeker's Armguard, reducing its effectiveness early game, or miss out on stacking the tear.
Seeker's Armband provides up to 35 AP. 35 AP might seem like nothing to scoff at. It's actually laughable. This is why: 
Granting 40 AP and 300 mana with 10% CDR and 20% of max mana regenerated per level, this item is almost required for any mana mage to function with the Mana nerfs.
http://leagueoflegends.wikia.com/wiki/Lost_Chapter
AP mages that rely on mana can't actually USE the spells they have. Either they can, and they begin to become mana starved, or they lose out on cs/wave management/harassment damage, and this is the problem.
Zed can out-harass most mages now, simply because mages will run out of steam eventually.
Armband requires 5 whole waves of cs (not including siege minions) to actually reach its full potential, while Lost Chapter works right off the bat, and provides helpful assistance the LOWER the level you are, by refreshing mana more often, compared to Seeker's, which only helps once you've reached the end goal of 30 cs.
By adding a cloth armor, the gold cost becomes 1600 gold
Compare that to Seeker's Armband + Amplifying tome for 1535 gold.
Tear + Seeker's is 850 gold, so adding seekers to that is 1950 gold. You cannot feasibly compare this to either of the other without adding a kill + some cs. I will be ignoring this build path entirely, because no one is ever going to do this unless they get fed early through sheer dumb luck.
Seeker's armband path: 30-45 armor. 40-50 AP. No other bonuses.
Cloth Armor + Lost Chapter: 300 Mana, 20% max mana regen every level, 40 AP, 10% CDR and 15 armor.
Trading 15-30 armor, 65 additional gold, and 0-10 AP FOR 10% CDR, 300 Mana and mana regen every level is not even a question for most mages, simply because they can't use spells with the same reliability that Zed can with the nature of Energy's 10 regen per second. Base values are important, so 10 AP isn't actually that much.
The reason Lost Chapter is so damn important is that it's the only way mages can combat Zed's pattern with the same force. Being able to use spells faster through CDR and regaining loads of mana back per level is arguably the most important part of Lost Chapter. Cloth armor is just a band-aid to both negate lethality's damage bonus and to make it less effective, helping the mage survive in trades. 10 AP isn't too much considering you won't gain access to that until maybe an additional 3-4 minutes into getting the item, and AP ratios don't compare much to base values until mid-late game. You're gaining maybe 15-20 damage off of one rotation, assuming all spells land, and you're unable to use them as much OR as often.
In conclusion/TL;DR, Yes, Seeker's Armband allows a mana-based mage to survive. It does NOT allow them to trade, harass, and cs effectively. Zed's harassment through spells is simply too fast through W>E>Q / Q alone. A mage trades 15-30 armor, 65 additional gold, and 0-10 AP FOR 10% CDR, 300 Mana and mana regen every level by comparing these two builds: Seeker's Armband + Amplifying Tome, and Lost Chapter + Cloth Armor.
PS: If you want to go a step further, Seeker's Armband + Amplifying Tome + Mana Crystal vs Lost Chapter + Cloth Armor + Cloth Armor:
0-15 armor, 15 additional gold, and 10-30 AP FOR 10% CDR, 50 Mana and mana regen every level. CDR is so incredibly important for mages early on to stay toe to toe with Zed, and the reason that higher elo players don't struggle with Zed is both 1: They can utilize spells more effectively, thereby reducing the amount of trading Zed can do successfully, and 2: Tend to use more effective mid-laners that can roam, like Talon or Aurelion Sol, making "Lower Elo" Zed and "Higher Elo" Zed two completely different arguments.
A Mid laner in Lower Elo is one that gets fed THEN roams, a Mid laner in Higher Elo is one that roams THEN gets fed, while simultaneously getting 2 other people fed.
