Kevin Ford: On the theory of prime producing sieves, part 2 (NTWS 224)

Abstract: The closest thing to a general method for counting primes in a set is the method of Type I/II sums. This allows one to obtain an asymptotic formula (or perhaps a non-trivial lower bound) for the number of primes in the set, provided one has sufficiently good estimates for certain auxiliary sums.
Unfortunately what counts as 'sufficiently good' is poorly understood, as are the limits of this approach. In this talk I'll talk a new framework (joint with James Maynard) which allows us to prove various necessary and sufficient conditions, focusing on methods for constructing sets that satisfy the Type I and Type II bounds yet contain no primes. In particular, I will go into some detail about how to prove that a substantial 'Type II range' is necessary to deduce the existence of primes in a set.
Link to slides: drive.google.c...
Number Theory Web Seminar: www.ntwebsemin...
Original air date:
Thursday, October 17, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 18, 2024 (2am AEDT, 4am NZDT)