Yu-Chen Sun

On Halasz‘s theorem

The prime number theorem tells us the number of primes up to x is (1+o(1))x/log x. An equivalent form is that Σmu(n)=o(x), where mu is the Mobius function which is 1-bounded multiplicative. It is natural to study the properties of 1-bounded multiplicative functions f such that Σf(n)=o(x). In this talk, we will introduce Halasz theorem which asserts that if a 1-bound function doesn’t “pretend” to n^it, then Σf(n)=o(x).