Distribution of multiplicative functions and other topics in analytic number theory

乘法是算术运算中的主要运算之一。理解这个简单操作的精细性质会导致令人惊讶的深刻问题。例如，质数是整数（整数），不能写成两个其他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“积木”，因为任何整数都可以写成素数的乘积。了解素数在所有整数的集合中的分布，以及在整数的其他特殊子集中的分布，是数学中最受关注的问题之一。一个实用的方法来包装许多问题有关的素数，更一般地说，乘法，是通过使用乘法函数。这些函数是关于整数的乘法结构的。例如，6 = 2 x 3，所以如果f是一个乘法函数，我们向它输入6，2和3，输出f（6），f（2）和f（3）必须遵守相同的乘法定律：f（6）= f（2）x f（3）。通常，向乘法函数输入不同的整数会产生不可预测的输出。但是，如果我们将这些输出平均到许多输入上，那么不规则性就会被平滑，我们可以开始提出各种问题。也许最重要的一个问题是这个平均值有多大。这个问题变得特别有趣和困难时，我们的函数不承担只有积极的或只有消极的价值，而是，它之间来回跳跃的积极和消极的价值。那么，我们如何知道这些相反符号的值是否相互抵消呢？理解这种味道的问题是我研究的主要部分。素数是如此的基本，以至于有可能在你面前最令人惊讶的地方找到它们。椭圆曲线既不是曲线也不是椭圆。相反，它们是具有几个显著特性的甜甜圈状表面。特别是，可以在它们的点上定义一个操作，就像我们在整数上定义乘法一样。研究这种运算可以产生什么样的对称性，自然地使我们想到了关于算术级数中素数分布的问题（例如，在4，7，10中，...）。

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,...)