How to get through the rigorous phase of PhD?
Take a (short) vacation immediately.
By working 15-16 hours a day you are making yourself too exhausted and tired to be effective. That is also probably why you make mistakes. You should take a break that is long enough to reset, and when you come back aim at reasonable working hours (depends on a person and circumstances but for most people this is 40-60hrs/week).
i spent on study i in a day from 15 to 16 hours I do study [...] My productivity is less these days as I try to avoid mistakes in my research. [...] I always try to prepare my best but fails these day quite frequently and making silly mistakes which are annoying [...]
You have already identified the problem. You're over-worked and under-productive. First, I would take a vacation over a long weekend. You need a break.
After your vacation, you need to better manage your time so you will have a sustainable work schedule. From your post, it sounds like you may need to build better time management skills. I've seen grad students waste time doing the following
1) Spending several hours reading 1 paper. Spend a set amount of time (like 30 minutes) reading each paper and note down what you don't understand. Join a reading group and bring your questions there.
2) Taking too many classes at once to "get them out of the way." Sometimes your advisor will push you to do this. If this is the case, stand firm, and set some other goals with them.
3) Constantly rework a paper, or otherwise pour time into a fruitless research topic. Part of your post mentioned you feel behind on the number of papers published. Bring this up with your advisor and focus on upping that number. This may mean publishing shorter papers or publishing in less prestigious journals. Once you zero in on a good research topic, it'll be much easier to turn out high quality research papers.
Your experience does not sound incredibly different from my own. (I finished my math PhD less than a year ago.) That said, I can tell you about a few things I did that made a big difference for me during my thesis years.
Deliberately set boundaries for your work. For example, I never did anything academic while eating, and I never worked on Fridays after 6:30 p.m. Those times were deliberately for my brain to rest.
Commit to doing something non-academic to force you to get away from your desk. I volunteered with a local church for a few hours a week. It was only a small commitment, but I found that I benefited greatly from the time away from my work.
It's a cliche, but exercise also made a big difference for me. You don't have to be super intense to get the benefits--I would usually run or lift for about 45 minutes three or four times a week. Take advantage of your university gym if you can!
This is my most important point: Do NOT allow yourself to feel guilty for spending time away from your work. You've been in mathematics long enough now to know that revisiting a problem after a short break can grant you new insights. Just like the people in other answers have said, grinding out 16 hour days can lead to a significant drop in productivity, so you should aspire to have more productive work with less hours spent--that begins with the implementation of breaks.