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UNIVERSITY of CALIFORNIA Los Angeles UNIVERSITY OF CALIFORNIA Los Angeles Dynamical Methods for the Sarnak and Chowla Conjectures A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Redmond McNamara 2021 © Copyright by Redmond McNamara 2021 ABSTRACT OF THE DISSERTATION Dynamical Methods for the Sarnak and Chowla Conjectures by Redmond McNamara Doctor of Philosophy in Mathematics University of California, Los Angeles, 2021 Professor Terence Tao, Chair This thesis concerns the Liouville function, the prime number theorem, the Erd}osdiscrepancy 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 κ − 1-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with O(nt−") many words of length n where t = κ(κ+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}osdiscrepancy problem. In particular, we show that for any N and any sequence f of plus and minus ones, for some ii 1 P 484 −o(1) n ≤ N and d ≤ exp(N) that j i≤n f(id)j ≥ (log log N) . iii The dissertation of Redmond McNamara is approved. Rowan Killip Bill Duke Tim Austin Terence Tao, Committee Chair University of California, Los Angeles 2021 iv TABLE OF CONTENTS 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 2 Sarnak's Conjecture for Sequences of Almost Quadratic Word Growth : 9 2.1 Introduction to Chapter 1 . .9 2.1.1 Background and notation . 21 2.1.2 Acknowledgments . 28 2.2 Main Argument . 28 2.2.1 The Entropy Decrement Argument . 46 2.2.2 Nilsystems and Algebraic Structure . 54 2.3 Proof of Theorem 2.1.11 . 74 2.4 Frantzikinakis-Host and dynamical models . 75 2.5 Reduction to the completely multiplicative case . 79 3 A Dynamical Proof of the Prime Number Theorem ::::::::::::: 84 3.1 Introduction to Chapter 2 . 84 3.1.1 A comment on notation . 89 3.1.2 Acknowledgments . 90 3.2 Proof of the prime number theorem . 90 3.3 In what ways is this a dynamical proof? . 109 4 A Quantitative Erd}osDiscrepancy Theorem ::::::::::::::::: 118 4.1 Introduction to Chapter 3 . 118 4.2 Multiplicative Fourier Reduction . 125 v 4.3 The Nonpretentious Case . 130 4.4 The Pretentious Case . 135 References ::::::::::::::::::::::::::::::::::::::::: 151 vi ACKNOWLEDGMENTS I would like to thank Terence Tao for his help and guidance. I would also like to thank Tim Austin, Carlos Kenig and Amie Wilkinson for their generous assistance. Finally, I would like to thank my family, especially my parents Bruce McNamara and Daria Walsh for their love and support. vii VITA 2016 B.A. (Mathematics), University of Chicago 2018 M.A. (Mathematics), University of California, Los Angeles 2020 Beckenbach Fellowship, Department of Mathematics, University of Cali- fornia, Los Angeles PUBLICATIONS McNamara, Redmond, Sarnak's Conjecture for Sequences of Almost Quadratic Word Growth, Ergodic Theory and Dynamical Systems (2020). McNamara, Redmond, A Dynamical Proof of the Prime Number Theorem, Hardy Ramanujan Journal (to appear). viii CHAPTER 1 Introduction Arguably the oldest known algorithm for finding primes is the sieve of Eratosthenes, first recorded by Nicomachus in the second century A.D. but attributed to the third century B.C. mathematician. Up to cosmetic modifications, the algorithm works as follows. Let x be a natural number. Then write a list of all the numbers between x and 2x. Cross off all the multiples of 2, then all the multiples of 3, then all the multiples of 5 and so on. What remains p after everything else has been crossed off (say up to multiples of 2x) are the primes. This naturally gives a way to count primes. The number of primes between x and 2x is x minus the number of numbers divisible by 2, minus the number of numbers divisible by 3 minus the number of numbers divisible by 5 and so on. Except now we have twice subtracted off the number of numbers divisible by 6 because we subtracted them off once when we subtracted off the multiples of 2 and once when we subtracted off the multiples of 3. Similarly with 10, 15 et cetera. So we add on all the multiples of 6, all the multiples of 10, all the multiples of 15 and so on. But now again we have over counted the multiples of 30. Altogether, we get the formula # of primes between x and 2x = x − # of #s divisible by 2 − # of #s divisible by 3 − # of #s divisible by 5 − · · · + # of #s divisible by 6 + # of #s divisible by 10 + # of #s divisible by 15 + ··· − # of #s divisible by 30 − · · · 1 or to rewrite this using variables X # of primes between x and 2x = ± # of #s divisible by d : p d≤ 2x d squarefree How do you know whether you are supposed to add or subtract the multiples of d? The coefficient of the ( # of #s divisible by d ) is precisely (−1)# of prime factors of d. This is the Liouville function λ(d) = (−1)# of prime factors of d; and its central role in the sieve of Eratosthenes is the first of its many important roles in number theory. Another hint that the Liouville function might be an important function in the study of the primes is the identity ζ(2s) X λ(n) = ζ(s) ns n2N where ζ is the famous Riemann zeta function defined by X 1 ζ(s) = ns n2N −1 Y 1 = 1 − : ps p Therefore the famous zeros of the zeta function correspond to poles of the Dirichlet series for λ. Yet another hint that the Liouville function might be central to analytic number theory is given by the convolution formula Λ = λ ∗ log ∗ φ where φ(d2) = λ(d) if d is squarefree and 0 otherwise i.e. φ(d2) = µ(d) and Λ is the von Mangoldt function, which acts sort of like the normalized indicator function of the primes. All of this perhaps makes the following fact less surprising: the prime number theorem, x which states that the number of primes less than x is log x (1 + error) where the error term goes to 0 as x tends to infinity, is equivalent to the fact that lim n≤N λ(n) = 0; n!1 E 2 where by definition 1 X f(n) = f(n): En≤N N n≤N There are many interesting ways of thinking about this theorem, which in turn lead to their own interesting generalizations. First, we can think of lim n≤N λ(n) = 0 n!1 E as telling us that λ(n) = +1 roughly 50% of the time and λ(n) = −1 roughly 50% of the time. One might naturally then ask what about (λ(n); λ(n+1))? Is that (+1; +1), (+1; −1), (−1; +1) and (−1; −1) with equal probability as well? What about (λ(n); λ(n+1); λ(n+2))? It turns out that the answer to this question is still unknown and would represent major progress in analytic number theory. In fact, this is a famous conjecture. To state it precisely, define a word of length k of a sequence f as a string of k consecutive values of f i.e. is a word of f if there exists n such that f(n + i) = i for all i ≤ k. Then Chowla's conjecture states Conjecture (Chowla). All 2k possible words of length k of the Liouville function occur with equal probability. This is open. In fact, we do not even know whether all 2k possible words of length k occur at all. Previously [Hil86a] showed that all 8 words of length 3 occur infinitely often. [MRT16] showed all 8 words of length 3 occur with positive density. [Tao16b] proved that all 4 words of length 2 occur with equal probability if one uses logarithmic weights rather than the normal uniform weights. (We will define logarithmic weights later on). [TT17b] proved that all 16 sign patterns of length 4 occur with positive density using an argument communicated to them by Matom¨akiand Sawin. [TT17b] showed that all 8 words of length 3 occur with equal probability again using logarithmic weights. [TT17b] also showed the 3 number of words of length k is at least 2k + 8 for k ≥ 4. [FH18b] showed that the number of words is super linear. In this thesis (see Chapter 2), based on work in [McN18], we show Theorem. There is a constant c such that at least ck2 many words of length k occur with positive upper logarithmic density. Since [McN18] was published, it was proved in [MRT+] that actually super polynomially many words occur, but not necessarily with positive density. Another way to generalize lim n≤N λ(n) = 0 n!1 E is to ask about multiple correlations. Is it true that lim n≤N λ(n)λ(n + 1) = 0? n!1 E What about the more general question lim n≤N λ(n + h1)λ(n + h2) ··· λ(h + hk) = 0 (1.1) n!1 E where none of the hi's are equal? Clearly this would follow from the Chowla conjecture as stated earlier.
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