Distribution of multiplicative functions and other topics in analytic number theory

乘法是主要的算术运算之一。理解这一简单操作的精细性质会带来令人惊讶的深刻问题。例如，素数是整数（整数），不能写成另外两个严格较小的整数的乘积。所以2 3 5 7 11…都是质数。这些特殊的整数是乘法的“基石”，因为任何整数都可以写成质数的乘积。理解质数在所有整数集合中的分布，以及在整数的其他特殊子集中的分布，是数学中最受欢迎的问题之一。

Multiplication is one of the main arithmetic operations. Understanding fine properties of this simple operation leads to surprisingly deep questions. For example, the prime numbers are integers (whole numbers) that cannot be written as product of two other, strictly smaller integers. So 2, 3, 5, 7, 11, ... are prime numbers. These special integers are the 'building blocks' of multiplication, since any integer can be written as the product of prime numbers. Understanding the distribution of prime numbers among the set of all integers, as well as among other special subsets of the integers, is one of the most sought questions in mathematics. A practical way of packaging many questions concerning the prime numbers and, more generally, multiplication, is by using multiplicative functions. These are functions that respect the multiplicative structure of the integers. For example, 6 = 2 x 3, so if f is a multiplicative function and we input 6, 2 and 3 to it, the outputs f(6), f(2) and f(3) will have to obey the same multiplicative law: f(6) = f(2) x f(3). Often, inputing different integers to a multiplicative function can produce outputs that vary unpredictably. But if we average these outputs over many inputs, then the irregularities are smoothened out and we can start asking various questions. Perhaps the single most important question is how big this mean value is. This problem becomes of particular interest and difficulty when our function does not assume only positive or only negative values but, instead, it jumps back and forth between positive and negative values. How do we know then if these values of opposite signs cancel each other? Understanding questions of this flavour is a major part of my research. Prime numbers are so fundamental that it is possible to find them in front of you in the most surprising places. Elliptic curves are neither curves nor elliptical. Rather, they are donut-like surfaces which possess several remarkable properties. In particular, it is possible to define an operation on their points, much like we define multiplication on the integers. Studying what symmetries this operation can give rise to, led us naturally to questions about the distribution of prime numbers inside an arithmetic progression (e.g. in 4,7,10,...)