I play as sivir all the time. I buy berserkers greaves, then manamune, then trinity force. Then i need attack speed and well speed so i buy 2 zeals. usually i dobt get to buy it but if i get the chance i get the bloddthirster. At the point of 2 zeals i am basicaly unstoppable. So yeah.

nice build but manamune sucks

what last guy said. Your E is all you need for those mana problems.

Several things that I have problems with in this guide.
First, manamune. Yes, Sivir is VERY mana hungry early on. You know how you solve this without costing a precious item slot?

Conserve mana. Don't even consider throwing out a Q unless it's for sure going to hit a champion WITHOUT hitting minions.

Once you get really good at Sivir's spell shield, you can even bait out opposing spells and simply shield them. There are also spells with a long animation or slow missile speed (Taric's stun, Caitlyn's ultimate immediately come to mind) that are very easily blocked, allowing you to gain some mana back.

Also, instead of attack speed glyphs, it's much MUCH more effective to grab the scaling Magic Resist per level glyphs to deal with mid to early-late game AP carry damage.

Finally, touching on item choices again, I 99% of the time rush Infinity Edge on Sivir because it is a RIDICULOUS source of damage if you do her ricochet combo (which you neglected to mention completely).
For those unaware, her W (ricochet) actually resets her swing timer. This means you can get three successive attacks out in about 2 seconds with NO attack speed items. You can achieve this by auto-attacking, pressing W after the auto attack animation launches her blade, forcing another auto-attack, and then auto-attacking again.

Why is this combo so potent? Her first strike of ricochet applies crit, lifesteal, and any other on hit effect. So you can potentially crit 3 times in about 2 seconds, which is HUGE if the enemy carry is already under half health. If they aren't then they are quickly forced out of lane to go heal or risk dying, as even with Soraka backing them up, your W and Q are on a much shorter cd than most heals (sans Sona, who doesn't have enough healing to sustain that amount of burst anyways)

I might edit this as I think of more.

Manamune on Sivir is good...in early game you have got many Manaproblems so Manamune is good to be able to harass your opponenents better

Just the Manamune is wrong, if you would go BT > IE > PD > LW it would be a basic Sivir build (due to your ult you can delay your PD and just keep your Zeal). But yeah, without the LW your autoattacks will never hit above the 500 damage mark =p

Good for new sivir players, though this build will never make it too anything near competitive.