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Arithmetic Structure and Dependent R...

Xu, Wenqiang,

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Arithmetic Structure and Dependent Randomness /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Arithmetic Structure and Dependent Randomness // Wenqiang Xu.
Author:	Xu, Wenqiang,
Description:	1 electronic resource (109 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 86-04, Section: B.
Contained By:	Dissertations Abstracts International86-04B.
Subject:	Multiplication & division. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31520332
ISBN:	9798342113380

Arithmetic Structure and Dependent Randomness /
Xu, Wenqiang,

Arithmetic Structure and Dependent Randomness /Wenqiang Xu. - 1 electronic resource (109 pages)

Source: Dissertations Abstracts International, Volume: 86-04, Section: B.

The thesis mainly focuses on the study of how probabilistic ideas and techniques interact with arithmetic structures. In analytic number theory, a lot of interesting arithmetic problems, such as the distribution of primes, and the Riemann Hypothesis are all connected to the study of multiplicative functions f, which is an arithmetic function defined on positive integers such that have multiplicative structure, i.e. f(mn) = f(m)f(n). Famous examples include M¨obius function µ(n), Liouville function λ(n) or n it for some t ∈ R.Understanding arithmetic information about these multiplicative functions is interesting and challenging. For example, a statement like for all large x, is equivalent to the Riemann hypothesis, which encoded the information about the prime distribution. This suggests that understanding partial sums of multiplicative functions is an important but also challenging task. An intuitive reason to believe (1.0.1) holds is that we have evidence that µ(n) is kind of behaves like a random sequence. For example, the prime number theorem shows that the partial sum is o(x). If the µ(n) is indeed like independent random variables Xn taking values from {1, −1} for all square-free n, then the classical result in probability shows that very likely the random sums should have square-root cancellation.Another parallel and important example is about the typical character sums. For example, if we consider the family of Dirichlet character mod r, then we would like to know that cancellation amongst the character sums (for various ranges of x and r)Here one might want to guess that χ(n) behaves like a random variable taking values uniformly distributed on the complex unit circle. This might lead to a square-root cancellation guess without any computation.In the last couple of decades, it turns out that many ideas from the probabilistic world have played an important role in the study of arithmetic objects. The area of studying questions on the intersections of arithmetic and probabilistic flavor is usually called probabilistic number theory. The thesis can be viewed as devoted to the study of this area. In particular, we focus on studying a probabilistic model, called random multiplicative functions (RMF). A Steinhaus random multiplicative function f(n) is a completely multiplicative function defined on positive integers such that f(p) are independently and uniformly distributed on the complex unit circle for all primes p. One may view it as a random model for arithmetic functions like Dirichlet characters χ(n) or nit. Another popular model is the Rademacher random multiplicative function f(n) which was first used by Wintner [109] as a random model for M¨obius function µ(n). Most of the time in this thesis, we focus on the Steinhaus case. The obvious dependence between random variables f(m) and f(n) whenever (m, n) ̸= 1 makes the study of random multiplicative functions intriguing. The intriguing structure here actually connects to the study of Gaussian multiplicative chaos, barrier methods, branching random walks, etc in the probabilistic literature.There are several main motivations and research directions for studying these probabilistic models and related number theoretical questions.

English

ISBN: 9798342113380Subjects--Topical Terms:

3680536
Multiplication & division.

Arithmetic Structure and Dependent Randomness /
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500 $a Advisors: Fox, Jacob; Soundararajan, Kannan Committee members: Conrey, Brian; Stacey, Stacey F.
502 $b Ph.D. $c Stanford University $d 2024.
520 $a The thesis mainly focuses on the study of how probabilistic ideas and techniques interact with arithmetic structures. In analytic number theory, a lot of interesting arithmetic problems, such as the distribution of primes, and the Riemann Hypothesis are all connected to the study of multiplicative functions f, which is an arithmetic function defined on positive integers such that have multiplicative structure, i.e. f(mn) = f(m)f(n). Famous examples include M¨obius function µ(n), Liouville function λ(n) or n it for some t ∈ R.Understanding arithmetic information about these multiplicative functions is interesting and challenging. For example, a statement like for all large x, is equivalent to the Riemann hypothesis, which encoded the information about the prime distribution. This suggests that understanding partial sums of multiplicative functions is an important but also challenging task. An intuitive reason to believe (1.0.1) holds is that we have evidence that µ(n) is kind of behaves like a random sequence. For example, the prime number theorem shows that the partial sum is o(x). If the µ(n) is indeed like independent random variables Xn taking values from {1, −1} for all square-free n, then the classical result in probability shows that very likely the random sums should have square-root cancellation.Another parallel and important example is about the typical character sums. For example, if we consider the family of Dirichlet character mod r, then we would like to know that cancellation amongst the character sums (for various ranges of x and r)Here one might want to guess that χ(n) behaves like a random variable taking values uniformly distributed on the complex unit circle. This might lead to a square-root cancellation guess without any computation.In the last couple of decades, it turns out that many ideas from the probabilistic world have played an important role in the study of arithmetic objects. The area of studying questions on the intersections of arithmetic and probabilistic flavor is usually called probabilistic number theory. The thesis can be viewed as devoted to the study of this area. In particular, we focus on studying a probabilistic model, called random multiplicative functions (RMF). A Steinhaus random multiplicative function f(n) is a completely multiplicative function defined on positive integers such that f(p) are independently and uniformly distributed on the complex unit circle for all primes p. One may view it as a random model for arithmetic functions like Dirichlet characters χ(n) or nit. Another popular model is the Rademacher random multiplicative function f(n) which was first used by Wintner [109] as a random model for M¨obius function µ(n). Most of the time in this thesis, we focus on the Steinhaus case. The obvious dependence between random variables f(m) and f(n) whenever (m, n) ̸= 1 makes the study of random multiplicative functions intriguing. The intriguing structure here actually connects to the study of Gaussian multiplicative chaos, barrier methods, branching random walks, etc in the probabilistic literature.There are several main motivations and research directions for studying these probabilistic models and related number theoretical questions.
546 $a English
590 $a School code: 0212
650 4 $a Multiplication & division. $3 3680536
650 4 $a Mathematical functions. $3 3564295
650 4 $a Mathematicians. $3 514781
650 4 $a Central limit theorem. $3 648367
650 4 $a Combinatorics. $3 895617
650 4 $a Mathematics. $3 515831
690 $a 0405
710 2 $a Stanford University. $e degree granting institution. $3 3765820
720 1 $a Fox, Jacob $e degree supervisor.
720 1 $a Soundararajan, Kannan $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 86-04B.
790 $a 0212
791 $a Ph.D.
792 $a 2024
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31520332