On Halász‘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 $\sum_{n\leq x} \mu(n)=o(x)$, where $\mu$ is the Möbius function which is $1$-bounded multiplicative. It is natural to study the properties of $1$-bounded multiplicative functions $f$ such that $\sum_{n\leq x} f(n)=o(x)$. In this talk, we will introduce Halász’s theorem, which asserts that if a $1$-bound function doesn’t “pretend” to be $n^{it}$, then $\sum_{n\leq x} f(n)=o(x)$, and we will give different proofs of this theorem from different perspectives.