Yes, it is important that all students register. You can register for the TCC courses by emailing graduate.studies@maths.ox.ac.uk
The course will be assessed by solutions to the problem sheets. In total, ≥12 questions must be answered correctly.
Should an answer be poor or wrong, it will not count. If an answer is written up clearly, but contains minor errors I may sent a comment back on what went wrong and invite you to submit an improved or corrected answer which can count. — It is like submitting a paper to a journal. Depending on the quality it will be accepted, asked for a revision, or rejected.
No, there won't be a problem sheet for each week.
You can submit your homework whenever you like. I have not imposed any sharp deadline, just answer in total twelve questions correctly well before Christmas. You may answer more than this to be on the save side.
You can email me pdf files. I will acknowledge within a few days whether the answers are correct.
Feedback is a very valuable contribution to the course. It is an assessment on the lecturer. Feedback lets the teacher know whether he does his job to your fullest satisfaction. Should there be any discrepancy from what you expect, the feedback is there to help getting things straight. Also in the happy situation when everything is perfect, it is good to know. There is a TCC feedback form for anonymous submission.
Can be asked at the lectures, send by email, or via the TCC feedback form.
The Riemann function, also known as the Gram series, R(u)=1+Σn∈N un ⁄ (n!nζ(n+1)), and its relation to the prime number theorem can be found on Wikipedia.
Notation: Usually people write R(x) for what I've called R(log x).
To simplify the lecture a bit and to focus on the spectral interpretation, I have dropped minor correction terms in Riemann's explicit formula. If correctly written with all terms included, Riemann's explicit formula reads π(x)=R(log x)−Σ&rhoR(ρlog x)−1/log x+(1/π)arctan(π/log x).
A proof can be found online, where you can also follow the references to