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属性会带来令人惊讶的深刻问题。例如，素数是不能写成另外两个严格较小的整数的乘积的整数(整数)。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，是数学中最热门的问题之一。包装许多与素数有关的问题，以及更广泛地说，关于乘法的实用方法是使用乘法函数。这些函数尊重整数的乘法结构。例如，6=2x3，因此，如果f是一个乘法函数，我们向其输入6、2和3，则输出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，将不同的整数输入到乘法函数可能会产生不可预测的不同输出。但如果我们将这些产出除以许多投入的平均值，那么违规现象就会被消除，我们就可以开始提出各种问题。或许最重要的一个问题是，这个平均值有多大。当我们的函数不只假设正值或负值，而是在正值和负值之间来回跳跃时，这个问题就变得特别有趣和困难。那么，我们如何知道这些相反星座的价值是否相互抵消呢？理解这种味道的问题是我研究的一个主要部分。质数是如此基本，以至于在最令人惊讶的地方，你可能会发现它们就在你面前。椭圆曲线既不是曲线，也不是椭圆。相反，它们是像甜甜圈一样的表面，具有几个显著的特性。特别是，可以在它们的点上定义运算，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，自然会让我们产生关于算术级数中素数分布的问题(例如，在4，7，10，...)

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,...)