UCLA

This thesis concerns the Liouville function, the prime number theorem, the Erd\H{o}s discrepancy problem and related topics. We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce this theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the $\kappa-1$-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\e})$ many words of length $n$ where $t = \kappa(\kappa+1)/2$. We prove a variant of the $1$-Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension $< 1$. We give a new proof of the prime number theorem. We show how this proof can be interpreted in a dynamical setting. Along the way we give a new and improved version of the entropy decrement argument. We give a quantitative version of the Erd\H{o}s discrepancy problem. In particular, we show that for any $N$ and any sequence $f$ of plus and minus ones, for some $n \leq N$ and $d \leq \exp(N)$ that $| \sum_{i \leq n} f(id) | \geq (\log \log N)^{\frac{1}{484} - o(1)}$.