Early game blue helps MASSIVELY. It has several benefits.
1) increases his damage output drastically due to cdr.
2)allows him to farm much better, If you are pro and can last hit no problem with his AA, then grats to you, me and my 1500 elo i can't.
3)Allows charging up tear much faster, improving his damage output for majority of the game..
4) allows him to push lane so he can gank.

Blue is amazing on ryze until he gets at very least glacial.

Ryze is one of the least dependent on blue. It definitely helps to get the 2nd spawn since Ryze will have a tear but it won't be charged enough to keep him going. The blue will certainly help him charge the tear and keep spewing spells. After his second Back, he should be good to go.

But even then, since Riot keeps spewing out ridiculously long ranged champs, when I play Ryze I feel like Dodgeball cept Ive got no ball.
So I wind up dodging everyone elses skillshots and hardly ever harass the enemy. So I wind up using my Q for last hitting purposes more than anything. So even with the blue, all I could do with it is waste it, or be really lazy with my last hitting using Q instead of autoattack.
He has no AoE so he can't really push the lane either to go roaming. So that's out as well. There isnt really a need for sustained DPS during laning for the most part so the CDR isn't all that useful even.

Prior to getting glacial shroud, which many Ryze's will wait to build until after getting tear, t1 boots, and a catalyst for survivability, ryze won't have much CDR. The CDR from blue will help a lot.

If Ryze is facing a champ that is pushing them into their tower a lot, blue is EXTREMELY helpful, because it allows Ryze to push the wave back without running oom.

I like getting blue even after I have my shroud (if I don't think anyone else needs it more). You can spam your skills to charge your tear (if you do that without blue, you go oom fast). You can clear waves very fast/push your lane. You can keep someone pinned on their tower so you can go gank. Etc.

Later on in the game, Ryze does not need blue. But it is still nice to have if you can get it. Don't go out of your way to grab it if one of your teammates can get it, but if you have an opportunity, and no one else on your team who can use it is around, definitely pick it up. The people who say Ryze has infinite mana at end-game are incorrect.

Q costs 70, W 140, E 120.

Q-W-Q-E-Q-R-Q-W-Q-E-Q
Is 6 Qs, 2 Ws, and 2 Es.
That is 420 + 280 + 240 = 960 mana.
So a full combo takes almost 1/4th of your mana.

How much mana regen do you have? Well, you have an archangels staff (if you have completed your full build) for 25 mp5. And thats it.
And if you haven't fully finished your build (which costs around 14.5 or 15.5 depending on which items you get), you only have 7 mp5 from your tear.

And then another 3 mp5 from masteries.

And at lvl 18, I believe Ryze's base mp5 is 10.8.

So before you have a full build, you get 20.8 mp5 (assuming you don't have mana regen runes), which is 250 mana per minute. So if you are out of your base for 4 minutes, that nets you another full combo.

So you can do about 4 full combos before running OOM with 4k mana, plus another full combo for every 4 minutes you are out of your base, late game.

Once you get archangels, you get 25mp5 instead of 7 from it, for an additional 18mp5. Now you get another full combo for every 2 minutes or so you are out of your base, instead of every 4 minutes.

If you are really good with him, no he isn't. Super late game when your CDR is maxed and your lvl 18 so can can spam non-stop, yes, you will want blue, but early-mid game it's not necessary. It is a huge help though.

Late game, with the mp/5 from blue and the healing from your ult, you shouldn't need to go back at all.

Maybe it would be better to put it this way:

Ryze is a great mid to choose if you have ANOTHER champ that really needs blue buff to do well. For example, Amumu is very reliant on bluebuff and having both him and Anivia on a team would not exactly be an ultra combo.

Opinions vary, but you can see that generally Ryze can benefit from Blue but is not nearly as dependent as many AP mids, nor does he become appreciably better with blue buff compared to champs that can abuse their longer range.
I mean, you just have to accept that Ryze isn't a pusher and spamming Q to push lanes isn't a great plan.

So, hopefully if you play Ryze you can either:
a) get the modest boost from taking blue.
OR
b) donate the blue to a more mana intensive champ like KogMaw or Miss Fortune or Amumu, jungle Fiddle, etc.

You base your argument on statement that we are discussing to be true or false...

Ryze's run 9/0/21 which gives them 10% CDR, on top of the 10% he gets for maxing his Q, and frozen heart being a core item giving him 20%. Once he has this, he shouldn't need blue buff, however taking the first 2~3 blue buffs makes his laning phase and charging his tears infinitely easier.

It still gives him infinite mana however late game, but it's not huge and can be easily given to someone else on your team that can use the CDR + Mana regen more.

Yes I think Ryze is blue dependant early game. The way I and most other people play Ryze is to build a tear first so having that blue early on really lets you just spam your abilities to last hit minions and stack up on that tear passive. But after that he really has a bunch of mana so he really doesn't NEED it but I mean the cool down stacking with your passive cool down is always nice

huh? I thought the question was: How dependent is Ryze on blue buff, relative to Fiddle/Anivia/etc.

Since that question is pretty much answered (its mildly to moderately important), I am trying to look at it in a different way. How can we utilize Ryze's relative independence on blue to best effect?

This is a naive assumption. Pushing a lane to the enemy's tower is a viable way of winning the lane, especially in mid where defending against ganks is easier.

If you're up against someone like Morgana as Ryze, you'll be constantly at your tower trying to last hit. Ryze is terrible at that. He's about the worst champion in the entire game at last hitting with auto-attacks, so you'll lose a ton of CS to your tower. And your tower will take a significant amount of damage because Ryze isn't good at clearing waves without ult. And all that pushing gives Morgana tons of time to recall and buy to get the item edge or to roam and gank. All while you're stuck trying desperately to CS under your tower, where you can't recall or you'll lose XP in addition to even more CS and tower health.

To counter that, you need to have the ability to counter-push. That means being able to spam spells for Ryze. This means that Ryze needs blue buff. Blue buff isn't a modest boost - it's huge for Ryze until end-game where his build is finished. It lets him push better, sustain better, harass better, do more sustained damage, and charge his Tear faster. If Ryze doesn't get blue buff and goes up against enemy AP mids that do, he stands a strong chance of losing mid hard and never making it to the end-game where Ryze shines.