Probabilistic Aspects of Dirichlet Series by Simon Lyons Department of Mathematics Imperial College London London SW7 2AZ United Kingdom Submitted to Imperial College London for the degree of Master of Philosophy 2010 Abstract We investigate and generalise some properties of a family of probability dis- tributions closely related to the Riemann zeta function. Random variables that have the property that divisibility by a set of distinct primes occurs as a set of independent events are characterised in terms of functions that are well known in number theory. We refer to random variables with this independence property as Khinchin random variables. In characterising the collection of Khinchin random variables, we make a connection between the probabilistic theory of discrete distributions and the number-theoretic concept of Dirichlet series. We outline some interesting correspondences between discrete probability distributions and arithmetic functions. A subset of the Khinchin random variables have infinitely divisible logarithms. We establish the necessity of a condition, already known to be sufficient, that ensures infinite divisibility. Some Khinchin random variables admit a multiplicative decomposition into a product of random prime numbers. The number of terms in such a product follows a Poisson distribution. We explore two instances of this decomposition: one related to the zeta distribution, and the other related to the so-called prime zeta function. We use the zeta distribution to derive known results from number theory via probabilistic methods, and provide a generalisation of the distribution for other unique factorisation domains. Acknowledgements I would like to thank Lane Hughston and Martijn Pistorius for introducing me to the material explored in this thesis, and Lane in particular for sharing insights derived from research he had previously conducted. Additional thanks to Don Blasius for his hospitality, and his comments on the nature of zeta functions, which added a new dimension to the nature of this work. I am grateful to Martijn Pistorius and Dorje Brody for their helpful com- ments on an earlier deaft of this thesis. Finally, thanks to Jorge Zubelli and the faculty at IMPA for allowing me to present my work at a fascinating and productive conference in Buzios. I would like to express my gratitude to the Fields Institute, Ontario, for funding to attend a research workshop in May-June 2010. The work reported herein was supported in part by an EPSRC DTA scholarship at Imperial College London. 3 Declaration The work presented in this thesis is my own. Simon Lyons, December 2010. 4 Contents 1 Introduction 6 1.1 Preliminary remarks . 6 1.2 Dirichlet series and the zeta function . 7 1.3 The zeta distribution . 10 2 Khinchin distributions 14 2.1 The factorisation property . 14 2.2 Prime Factors . 15 2.3 Characterisation of Khinchin distributions . 17 2.4 Construction of multiplicative arithmetic functions . 21 2.5 Changes of measure . 23 2.6 Examples of Khinchin distributions . 25 3 Infinite divisibility 32 3.1 Logarithms of Khinchin distributions . 32 3.2 Examples of Khinchin random variables with infinitely divisi- ble logarithms . 39 4 Density 42 4.1 Analytic and asymptotic density . 42 4.2 Examples from number theory . 45 5 Concluding remarks 50 6 Appendix 53 6.1 Miscellaneous observations . 53 6.1.1 The prime number theorem for arithmetic progressions 53 6.1.2 Gaussian integers . 54 6.2 Index of notation . 58 5 Section 1 Introduction 1.1 Preliminary remarks In this thesis, I explore properties of a family of probability distributions closely related to the Riemann zeta function. The theory is applied to a more general class of functions known as Dirichlet series. The first section of the report introduces the zeta function, Dirichlet series, and various properties of arithmetic functions. One can define an arithmetic function as a function f(n) that expresses some property of the integer n. We introduce, for each real s > 1, the zeta distribution, Zeta(s), and develop intuition about its behaviour. In Section two, we construct a class of probability distributions closely related to Dirichlet series, which we refer to as distributions of the Khinchin type. A random variable X that has a Khinchin distribution has the following property: if m and n are mutually prime positive numbers, then the events fm divides Xg and fn divides Xg are independent. In Theorem 3.6, we provide a necessary and sufficient condition for a probability distribution to be of the Khinchin type. In Section three, we find a condition on a Dirichlet series that holds if and only if the logarithm of the associated Khinchin random variable is infinitely divisible. We use infinite divisibility of the logarithm of a zeta random variable to find a new representation of the zeta distribution in terms of random prime numbers, presented in Example 3.2 in Section three. Section four links the probabilistic theory developed in the previous chap- ters to various aspects of classical number theory. The limiting case of the zeta distribution as s approaches unity is not a bona-fide probability distri- bution. Nevertheless, one can use the limiting case to study number theoretic concepts known as densities. In Example 4.1, we prove a weak version of the 6 prime number theorem. We examine an analogue of the Erd¨os-Kactheorem that applies to the zeta distribution. Roughly speaking, the Erd¨os-Kacthe- orem states that the number of distinct prime factors of a large number n behaves like a sample from a normal distribution with mean and variance log(log(n)). In the appendix, we sketch some miscellaneous ideas that do not fit into the natural flow of the main body of the thesis. This section is intended to be read as a heuristic guide to possible avenues for further development of the theory. We choose not to sacrifice clarity for the sake of brevity. Some calculations may be laid out in rather more detail than is strictly necessary. Our aim is to make comprehension of the subject as straightforward as possible. While it is possible that some results in this thesis may already be known, all unattributed calculations are due to the author, and are original to the best of my knowledge. A brief summary of notation is included in the appendix. 1.2 Dirichlet series and the zeta function The Riemann zeta function is a central object of study in analytic number theory. Euler demonstrated, for positive integral values of s greater than unity, that the sum 1 X 1 ζ(s) := < 1; (1.1) ns n=1 converges, and that the sum diverges when s = 1. The domain of the zeta function was extended by Chebychev from positive integral values of s > 1 to real values of s > 1. Riemann showed that the zeta function admits an analytic continuation as a holomorphic function on the complex plane for all s 2 C except for a simple pole at s = 1. The literature on the zeta function is vast, and no attempt at a survey will be made here. We mention Whittaker and Watson [45], Titchmarsh [44], and Ivi´c[27], for example, as being well-known accounts of the properties of ζ(s). See Edwards [16] for historical information about the zeta function and an English translation of Riemann's original memoir on the subject. A crucial feature of the zeta function is that for Re(s) > 1 it can be expressed as a product over terms involving the prime numbers. This is a consequence of the fundamental theorem of arithmetic, which states that each integer admits a unique factorisation into primes. In particular, the 7 following identity holds: Y −1 ζ(s) = 1 − p−s ; Re(s) > 1; (1.2) p where p is understood to run over the set of all primes. The plausibility of (1.2) can readily be seen by expanding each term in the product as a power series, and formally multiplying the terms out. See, for example, Apostol [2, Chapter 11] for more on Euler products. One can generalise the notion of a zeta function in the following way. Suppose we have some function a : N ! C defined on the positive integers. We refer to a(n) as an arithmetic function. We form the series 1 X a(n) A(s) := ; (1.3) ns n=1 assuming that the series converges for some s 2 C. We refer to A(s) as a Dirichlet series. A(s) is also known as the generating function of a(n). Apostol [2, p. 233] shows that if a Dirichlet series converges absolutely for a complex number s0 = σ0 + it0 then it converges absolutely for all s 2 C sat- isfying Re(s) > σ0. The same principle applies for conditional convergence. Unless otherwise noted, we shall assume in what follows that Dirichlet se- ries are absolutely convergent for all s with real part greater than some real number σ0. Multiplication of Dirichlet series is related to a well-known binary opera- tion on arithmetic functions. If F (s) and G(s) are Dirichlet series, then the product 1 ! 1 ! X f(n) X g(n) F (s)G(s) = (1.4) ns ns n=1 n=1 is another Dirichlet series, which we shall call H(s). By multiplying out the terms in (1.4), one can see that f(2)g(1) f(1)g(2) f(3)g(1) F (s)G(s) =f(1)g(1) + + + 2s 2s 3s f(1)g(3) f(4)g(1) f(2)g(2) f(1)g(4) + + + + + ::: (1.5) 3s 4s 4s 4s Grouping terms with the same denominator, one observes that 1 X h(n) H(s) = ; (1.6) ns n=1 8 where X n h(n) = f(d)g : (1.7) d d divides n This operation on the functions f(n) and g(n) is known as Dirichlet multi- plication, or Dirichlet convolution.