Topics in multiplicative and probabilistic number theory

乘法是主要的算术运算之一。了解它的优良特性会带来令人惊讶的深刻问题，我的研究计划很大程度上是受到这些问题的推动。乘性对象通常具有混沌的、类似随机的行为，因此从概率的角度研究它们是很自然的。素数是类随机乘性对象的一个重要例子。这些是整数(整数)，不能写成另外两个严格更小的整数的乘积。因此，2、3、5、7和11是它们中的头几个。素数是乘法的“积木”，因为任何整数都可以写成其中一些的乘积。数值计算很快揭示了素数的混沌本质，它们之间没有明显的结构。这就是为什么了解素数如何在所有整数中分布是出了名的困难的主要原因之一。*包装许多关于素数的问题的一个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=2 x 3，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，从而提供更多“混沌乘法对象”的例子。出于这个原因，我们以概率的方式接近它们，并对它们进行平均研究。了解这些平均值的性质是这项建议的主要目标之一。*素数具有非常简单的乘法结构，没有非平凡的因子。对于大多数其他整数来说，情况完全不同。以120为例：它的除数是数字1、2、3、4、5、6、8、10、12、15、20、24、30、40、60、120。整数因子的分布是这一提议的关键部分。整数的除数可以非常集中在某些点周围，从而形成大块。我特别感兴趣的是，对于一个“随机选择的整数”，这些块可以变得多大。*我的研究建议的最后一个组成部分是关于无理数的有理逼近。给出一组分母，我想知道“大多数”无理数是否可以用这些分母的分数来近似。据推测，如果我们的分母集“足够大”，答案是肯定的。事实证明，最困难的情况是分母有非常特定大小的公约数。因此，我们得出了另一种需要研究的乘法结构。

Multiplication is one of the main arithmetic operations. Understanding its fine properties leads to surprisingly deep questions, and my research program is largely motivated by them. Multiplicative objects often possess a chaotic, random-like behaviour, so that it is natural to study them from a probabilistic point of view.******An important example of a random-like multiplicative object are the primes. These are integers (whole numbers) that cannot be written as product of two other strictly smaller integers. So 2, 3, 5, 7 and 11 are the first few of them. Primes are the "building blocks" of multiplication, since any integer can be written as the product of some of them. Numerical calculations quickly reveal the chaotic nature of primes, with no apparent structure among them. This is one of the main reasons why understanding how primes are distributed among all integers is notoriously hard.******A practical way of packaging many questions about primes is to use 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) must 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, thus supplying more examples of "chaotic multiplicative objects". For this reason, we approach them probabilistically and study them on average. Understanding the nature of these averages is one of the main goals of this proposal.******Prime numbers have very simple multiplicative structure, with no non-trivial divisors. The situation is completely different for most other integers. Take, for example, 120: its divisors are the numbers 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. The distribution of the divisors of integers is a key part of this proposal. The divisors of an integer can get very concentrated around certain points, thus forming large clumps. I am particularly interested in understanding how big these clumps can get for a "randomly chosen integer".******The final component of my research proposal concerns rational approximations of irrational numbers. Given a set of denominators, I want to understand whether "most" irrational numbers can be approximated using fractions with these denominators. It is conjectured that the answer is yes if our set of denominators is "large enough". As it turns out, the hardest case is when the denominators have common divisors of very specific size. We thus arrive to another kind of multiplicative structure that needs to be studied.