Let's start again. The OP asks whether this question (which can be viewed at Mathematics) can be asked here, again. (There are two questions that OP asked at MO main that concern the function the OP is interested in, which appears in a paper of Connes, but these questions were closed and deleted.)
First, there is a question ban which we moderators don't have control over: this is due to a Stack Exchange algorithm running in the background. So for the time being, no, this or any question can't be asked until the question ban is lifted by the software. I don't know when that might be. But generally speaking the time period can be shortened by doing good things for the community such as giving good answers to questions.
Then the question becomes: can OP ask the question once the ban is lifted? Answer: the community will look upon it poorly if the OP tries to re-ask a question that was closed and deleted. In fact, question repetition is considered a site violation, and could incur a penalty such as an account suspension, especially if the OP has been made aware of policy and chooses to ignore it anyway. Instead of repeating a question that was closed, what should be done is improve the question that was closed. (But see the advice given below.) Particularly, the OP can undelete the self-deleted question Possible New way to prime number theorem: [sic], and then edit it in the hope that the community votes to re-open the question.
So far what I've done is recite site policy. As a practical matter, getting the community to re-open a closed question is not always easy. My impression is that the question as it currently stands (which asked "Is there any way we can prove prime number theorem using above series ?" [sic]) was not well-researched by the OP and is vague, open-ended. This type of question is not a good MO question. It would be close to impossible to prove that the PNT can't be proved using Connes' function, because one doesn't have access to the space of all possible proofs; on the other hand no reason is given for suspecting such a proof exists, even in outline form. To me it looks like not much more than a vague hope.
If the OP were to provide compelling reasons for why he/she thinks such a proof is in the offing, then it could be a different story. But I think that would require intensive research and investigation from the OP beforehand, before it gets to that stage. That's how it usually is at MO.
I have looked over some of the discussion and comments both here and at Mathematics Stack Exchange. The OP says he/she is just barely older than a teenager. In view of this, I am going to offer some unsolicited advice, with no disrespect intended. It's this: don't edit a question over and over and over in the hope it will one day be accepted. A well-crafted question is something thought over and worked out in advance, and usually reflects prior experience and immersion in the subject matter; normally, it will require little to no revision once it has been posted. If it requires many revisions, then it wasn't a well-crafted question to begin with. If you carry out many revisions, then it will likely become annoying to others as the question is repeatedly bumped in the effort to straighten it out.
My advice to young people who are relatively new to research is: don't be in a hurry or ambitious to get a question accepted at MO. As an end, it's not important. What is important is to let a question ripen and mature through the hard work you have put into it.
Look around at other well-received questions in the areas that interest you. Often times it will be the case that the OP just needed a nudge in the right direction because the OP's mind has been properly prepared through all the effort made in advance. On the other hand, I see hundreds and hundreds of questions coming from people who haven't put in the necessary work. Usually the answers will be wasted on them because they're not ready to receive the message, and they may founder for years this way, wasting time. Think hard, go deep, and then the questions proper for MO may emerge naturally.
